1.4 equations 47 1.7 unit review 85

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Contents

Unit 1 Calculator Usage and Equation Solving

Sections

1.1. Operations on the scientific calculator 9

1.2 Rounding off approximate numbers 29

1.3 Variables and formulas 39

1.4 Equations 47

1.5 Introduction to percent 61

1.6 Solving literal equations 77

1.7 Unit review 85

Unit 2 Topics in Algebra and Statistics

Sections

2.1 Coordinates and graphs 93

2.2 Slope of a line 125

2.3 Interpreting and finding equations of lines 143

2.4 Frequency distributions and graphs 165

2.5 Measures of central tendency 179

2.6 Measures of dispersion 197

2.7 Unit review 209

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Unit 3 Applications of Percent and Algebra

Sections

3.1 Proportions 223

3.2 Percent of difference 235

3.3 Applications of percent of difference 243

3.4 Simple interest 255

3.5 Installment loans 263

3.6 Compound interest 275

3.7 Unit review 283

Unit 4 Geometry and Units of Measurement

Sections

4.1 Introduction, definitions, angles 291

4.2 Triangles 307

4.3 Length, metric system 317

4.4 Area 329

4.5 Volume and capacity 343

4.6 Miscellaneous metric units and summary 357

4.7 Unit review 367

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Preface

The topics in MAT 155 are basic mathematical competencies that every two-year college graduate should possess. The contents closely follow recommendations from industry training personnel and from several departments at Greenville Technical College. Although it contains some algebra, this text does not assume that you have previously taken an algebra course. The objectives for each section are at the beginning of each section and inform you of what you should be able to do when you finish the section. The worked-out examples in each section reinforce concepts and are sometimes used to introduce new ideas. Read and work through them carefully. There are many sites on the internet where you may find additional explanations of many of the topics in the course. One good site is khanacademy.org. The problems at the end of each section should be worked to check your understanding of the material in the section. Answers to most of the homework problems may be found directly after the problems in each section. It is not good practice to look at the answer to a problem before trying to work it yourself. Once you have seen the answer, it is easy to deceive yourself into believing that you can solve the problem on your own even when you cannot. Problems involving most of the important concepts in a unit will be found in the unit review at the end of each unit. A summary of the important formulas of a unit will be found in the unit review. Thanks to diligent students who carefully read the text, worked all of the homework problems, and pointed out mistakes. Thanks also to faculty members who have spotted errors and made suggestions for improvements. Students

Please inform your instructor of any errors you find so that future versions may be corrected. We appreciate any comments or suggestions concerning readability or format of specific topics. In particular, if you need more homework problems, additional worked out examples, or different explanations of concepts in a particular section, let us know so that we may incorporate them into the next revision of the text. You may also submit suggestions to [email protected]. August, 2020

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UNIT 1

CALCULATOR USAGE AND EQUATION SOLVING

Unit 1 Calculator Usage and Equation Solving Section 1.1 Operations on the scientific calculator

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Objectives for this section Perform calculations on scientific calculator involving arithmetic operations, parentheses, exponents, negative numbers, square roots, scientific notation Change numbers from scientific to ordinary notation Understand order of operations

Order of Operations and Parentheses

When a computation such as 2 + 3 x 5 is to be performed, the order of operations must be clear. The expression could mean to add 2 + 3, and then multiply the result by 5 to get the answer 25; or it could mean to multiply 3 and 5, and then add 2 to get the answer 17. The correct answer is 17. Using parentheses to indicate what is to be done first, the expression means 2 + (3 x 5) rather than (2+3) x 5. There is a mathematical convention which says that if no parentheses appear, do all multiplications and divisions first, followed by all additions and subtractions. This is the order of operations followed by most calculators. Calculators often

display an asterisk as the symbol for multiplication. The computation of 2 + 5 x 3 - 14 7 using most calculators would yield the result 2 + 15 - 2 or 15, since the calculator would perform both the multiplication and the division before it did either the addition or the subtraction. If you wished to force the calculator to add 2 + 5 before it did anything else, then you could enter the following:

This would give the result 7 x 3 - 2 or a final answer of 19. There are two parentheses keys on your calculator, a left parenthesis and a right parenthesis. When you use parentheses, you must use both of these keys. To evaluate the expression 7 x (2 + 8) enter the following:

Most expressions are entered just as they look on paper.


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Parentheses must be used when performing division of complicated expressions such

as

7 9

6 2. Obviously, the answer to this calculation is 16 8=2 . If you enter the

following into your calculator, you will get the answer 10.5.

This is because the calculator does the division 9 ÷ 6 before it does either of the additions. The result is 7 + 1.5 + 2 which is 10.5. You should enter the problem on your calculator as follows:

Enclosing the numerator (top) and the denominator (bottom) of the fraction in parentheses will force the calculator to calculate each of them separately before it performs the division. An alternative method uses the unique features of the TI-30XS MultiView calculator.

Using the key allows you to enter the problem as a fraction using the following keystrokes:

Parentheses may also indicate multiplication. The numeral 23 simply means twenty three, but the expression (2)(3) or 2(3) or (2)3 means 2 x 3 even if the times symbol is omitted. The calculator recommended for this course will perform multiplication when parentheses are used. Calculating with Zero

any number + 0 = that number that is, 5 + 0 = 5 any number - 0 = that number that is, 8 - 0 = 8 any number x 0 = 0 that is, 6 x 0 = 0

0 / any number except 0 = 0 that is, 0 13 = 0 However, any number divided by 0 is undefined!

Hence, 9 0 or 9/0 cannot be calculated (calculator gives error message).


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Negative Numbers

Some calculations require the use of negative numbers. To enter a negative number into your calculator, use the negative key (-) followed by the number. The subtraction or minus key is never used to enter negative numbers. For example, to multiply -3 by -4 on the calculator, enter

to obtain the correct answer of 12. Note that entering a single multiplication symbol is quicker than enclosing a number in parentheses. Notice that the calculator is programmed to handle such things as “a negative number times a negative number gives a positive number.” If you have forgotten (or never knew) the rules for working with negative numbers, the calculator will take care of them if you enter negative numbers properly. The following rules apply to operations involving negative numbers and are built into your calculator: Add: If the two numbers have the same sign, the answer has that

sign. If the two numbers have opposite signs, eliminate the sign, subtract the two numbers, and give the result the same sign as the larger number had prior to subtraction.

Subtract: Change the sign of the number being subtracted and then add. Multiply or Divide: Multiply or divide the numbers without minus signs and then

make the answer negative if the signs on the two original numbers aren’t the same.

Exponents

Your scientific calculator can also compute with exponents. You may remember that 23 means (2)(2)(2) or that 54 means (5)(5)(5)(5). The small upper number is called an exponent or a power. (It is called a superscript on word processors.) It tells you how many times to multiply the lower number (called the base) times itself. The key on

the calculator that takes numbers to powers is and must be entered between the number and the exponent.


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Entering calculates 5 to the 4th power, which is (5)(5)(5)(5) = 625. Negative Exponents

A negative exponent means to take the reciprocal of the base number and then raise

the base to the positive power. The meaning of 5-4 is 4

1 1 or

6255 or. The calculator

keystrokes to do the preceding calculation would be:

. The decimal answer would be 0.0016. An

alternative method would be to use the key. In other words, a negative

exponent means to divide as in the following example: 3

3

1 110 0.001

100010.

Zero Exponents

Any number, except zero, taken to the zero power is 1. For example, 50 = 1, 100 = 1, (-2)0 = 1, etc. The only exception to this pattern is 00 which has no meaning. Try some of these on your calculator. The second (2nd) power of a number is often called the square of the number. That is, 52 may be called either "5 to the 2nd power" or "5 squared." Similarly, the third (3rd) power of a number is often called the cube of the number. Thus, 53 may be called either "5 to the 3rd power" or "5 cubed." Taking Negative Numbers to Powers

The number (-5)4 is not the same as -54. The first expression means to multiply -5 times itself four times. The second expression means multiply 5 times itself four times and then multiply by -1. The first answer is 625, while the second answer is -625.


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The order of operations is as follows:

1. Do all work inside parentheses first.

2. Next, do all exponents.

3. Perform all multiplications and divisions. (Work from left to right.)

4. Last, perform all additions and subtractions. (Work from left to right.)

Memory Aid: PEMDAS (Parentheses, Exponents, Multiplications/ Divisions,

Additions/Subtractions) “Please Excuse My Dear Aunt Sally.” Example 1. Calculate 3 + 5 x (2 + 43).

Answer: 3 + 5 x (2 + 64) Taking the number inside parentheses to the power

3 + 5 x 66 Adding the numbers within the parentheses

3 + 330 Multiplication before addition

333 The final answer

The problem could be entered into the calculator as

Note: You will need to arrow out of the exponent before you close the parenthesis. Scientific Notation

You may also use exponents to write very large or very small numbers in a form called scientific notation. Scientific notation uses a number between one and ten and a

power of 10 to express a number such as 560,000,000,000 in the form 115.6 x 10 .

Multiplying 5.6 times 1110 would move the decimal eleven places to the right and create the number 560,000,000,000.


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Example 2. The mass of the earth is approximately 5,980,000,000,000,000,000,000,000,000 grams. What is the scientific notation for this number?

Answer: . 275 98 x 10 because the decimal in 5.98 must be moved 27 places to the right to recreate the original number.

Example 3. Express 0.000 000 000 000 078 5 in scientific notation.

Answer: . 147 85 x 10 because multiplying by 1410 is the same as dividing by 10 fourteen times and would move the decimal fourteen places to the left. This would recreate the original number: 0.000 000 000 000 078 5.

To find the power of ten when changing a number from ordinary notation to scientific notation, starting from the left, make a mark after the first non-zero digit of the number and count the number of digits between this mark and the decimal. (The decimal is understood to be at the end of a whole number.) The result will be the exponent; however, you will have to supply the minus sign when needed for numbers less than one. Generally, entering a very long number into your calculator and pressing the “enter” key will automatically change the number into scientific notation. To change a number from scientific to ordinary notation, simply move the decimal the same number of places as the power of ten, to the right if the power is positive, to the left if the power is negative. Scientific calculators will automatically express the answer to a computation in scientific notation if the answer is too long to display on the screen. If you wish to have your answers displayed using scientific notation, you may accomplish this on the TI-30XS MultiView in the following manner:

1. Press mode

2. Use the arrow keys to highlight SCI and then press Enter

3. Press (quit) to get the display screen. Note that SCI is at the top of the screen.

The number previously on the screen will now be in scientific notation.


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key

This key is used to access the operations or functions written above the keys. Press

the key and then the key below the needed function. Do not press the key and another key at the same time.

For example: To turn the calculator off, press and release the key. Then press

the key. If you wish for your answers to be displayed in ordinary notation, you would need to make sure the calculator is in NORM mode by following the steps below. The NORM setting on this calculator will change a number from scientific notation into ordinary notation unless the number is too large to fit the display.

1. Press

2. Use the arrow keys to highlight NORM

3. Press

4. Press

If you need to enter into your calculator a number which is already in scientific notation, observe the sequence of keystrokes for the following example:

To enter 98.7 x 10 , the keystrokes are:

The result that will appear on the entry screen is 98.7 10 , if you are in scientific mode, or 8700000000 if you are in normal mode.


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The key combines the multiplication key , the number ten (10) and the

exponentiation key . You may enter 116.4 x 10 by using the following sequence

of keystrokes: Example 4. Calculation of the time it takes light to travel to the earth from the sun

involves the following calculation:

7

7

9.3 10

1.1 10 minutes.

To calculate this quantity on your calculator, one method is to enter

The result is . 08 454545455 10 .

An alternative method is to use the key as follows:

Example 5. Multiplication of 232.3 10 by 344.5 x 10 may be done by entering the following:

The result is . 581 035 10 .


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Example 6. Calculate 114.98 10 .

More Operations on the TI-30XS MultiView Calculator

Squaring Key

Entering gives the result 25 when the calculator is in NORM mode. In other words, this is just a fast way to take 5 to the second power. The

key gives the same result as

Example 7. Calculate 28.5 .

Answer: The result is 72.25.

Square Root Key

To find 25 on the calculator enter . The

number 5 is called the square root of 25. The square root of 25 is the number which

gives 25 when it is squared. 25 25

Example 8. Find the square root of 6.25.

Answer: 6.25 2.5 .


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Example 9. Calculate 6 30 .

Answer: 6 30 6 .


Answer: 41

If you wish to have a decimal answer, you will need to use the convert key .

Note: The feature of the calculator will substitute the value of the last answer calculated.


The calculator knows that the minus key and the key are different. Use the minus key only for subtraction and the negative key only for negative numbers. The negative key is pressed before the number is keyed in.


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Example 12. Calculate 1 0.07/4 .

Note: You could have used the key.

Power or Exponential Key

To perform operations that involve an exponential expression, key in the base of the

expression, then the power key and then the power.

Example 13. Calculate 42 .


Example 15. Calculate 4

2

NOTE: For examples 14 and 15, the answers are different. Therefore, the placement of the parentheses does matter!!


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Fractions or

When the division key is used, the decimal value of the fraction will be

calculated. When is used, the screen will display the number in fraction form.

To enter a mixed number, use the following keystrokes:


5.

Alternatively,


47

.


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Example 20. Convert 30

7 to a mixed number.

Note that the function will convert a mixed number to an improper fraction or an improper fraction to a mixed number.

Example 21. Calculate

7.1 6.5 11

5.7 1.12.


(52 15)0.05

152

.


22


48

25

7.25 10

2.5 10.


15

8

6.25 10

2.5 10.


17 17

9 9

3.5 10 2.1 10

2.6 10 3 10.

Unit 1 Calculator Usage and Equation Solving Section 1.1 Problems

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For problems 1 – 5, calculate and round your answer to two decimal places:

1. (5.88)(6.27) (4.16)(2.74)

2.

3.47 5.28 4.61

2.22(1.75 2.36)

3.

3.47 5.28 4.61

2.22 1.75 2.36

4.

3.47 5.28 4.61

2.22 1.75 2.36

5. 3.7 4.9 5.3 2.6

6. Calculate 5 12 8 2





11. Change 56,000,000,000,000,000 to scientific notation.

12. Change 0.000 000 000 000 000 006 89 to scientific notation.

13. Change 4,956,000,000,000,000,000,000 to scientific notation.

14. Change 0.000 000 000 000 000 948 3 to scientific notation.

15. Change 123.86 10 to standard notation.



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18. Calculate 9 123.86 10 4.46 10 and leave answer in scientific notation.

19. Calculate 15 205.28 10 2.88 10 and put answer in scientific notation.

20. Calculate 0.000 000 000 000 000 567 0.000 000 000 000 000 004 2 and leave answer in scientific notation.

21. Calculate 380,000,000,000,000,000,000 45,500,000,000,000,000,000 and

leave answer in scientific notation.

22. The U.S. National Center for Education Statistics gives 63.24 10 as the projected number of elementary and secondary teachers for the year 2050. Write this number in standard notation.

23. The population of the U.S. in 1994 was 82.603 10 . The land area of the U.S.

is 63.536 10 square miles. Find the number of people per square mile in the U.S. in 1994.

24. The mass of an oxygen atom is 0.000 000 000 000 000 000 000 013 g. Write this number in scientific notation.

25. A television signal travels 1 mile in 65.4 10 second. Write this number in standard notation.

26. If the national debt is about 127.09 10 dollars, and there are approximately 292 million citizens, how much would every citizen have to pay to completely eliminate the national debt?

27. Mars is about 141.3 million miles from the sun. Express this distance in

scientific notation. 28. About 65 billion aluminum drink cans are used each year. Write this number

using standard notation and then write it using scientific notation.

29. Calculate 214 .


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30. Calculate 2

12 .

31. Calculate 216 .

32. Calculate 423 .

33. Calculate 317 .

34. Calculate 351.07 and round your answer to 3 decimal places.

35. Calculate 27 4 8 4 20 .

36. Calculate 3

8 12 5 8 .

37. Calculate 3 32 5 .

38. Calculate 3

2 5 .

39. Calculate 4

3 .

40. Calculate 43 .

41. Calculate 4 3

7 2 8 2 20( 2) .

42. Calculate 25 16 .

43. Calculate 25 16 .

44. Calculate 16 (4)(2)( 5) and round your answer to 2 decimal places.

45. Calculate 2( 3) (4)(2)( 5) .


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46. Calculate 2

3 (4)( 1)( 12) .

47. Calculate 1 1 (4)( 6)

2.

48. Calculate 18 2(12 4 2 7) .

49. Calculate 2 212 3(6 2 5 9 3) .

50. Calculate 2 2 218 25 3 7 .

51. Calculate 2 28 12 and round your answer to 2 decimal places.

52. Calculate 2 217 23 and round your answer to 1 decimal place.



55. Calculate 316 3 2 5 and round your answer to 2 decimal places.

56. Calculate and round your answer to 2 decimal places. Give your answer in scientific notation.

13 12

5 5

2.88 10 4.76 10

3.55 10 4.72 10

57. Calculate 0

5 2 42.389 4.273 1.85 10 .

58. Calculate 4

12 137.86 10 4.49 10 and round your answer to 2 decimal

places. Give your answer in scientific notation.


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The following calculations involve money. Round your answers to the nearest cent.

59. Calculate

4 120.09

3000 14

.

60. Calculate

12 200.06

20,000 112

.

61. Calculate

365 250.075

15,000 1365

.

62. Calculate

(2 30)

40,000

0.051

2

.

63. Calculate

(12 30)

35,000

0.08251

12

.

64. Calculate

(365 14)

42,000

0.06751

365

.

65. Calculate

12 5

0.1412,000

12

0.141 1

12

.

66. Calculate

12 25

0.07150,000

12

0.071 1

12

.

67. Calculate 12 2.5

0.221600

12

0.221 1

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Unit 1 Calculator Usage and Equation Solving Section 1.1 Answers to selected problems

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1. 25.47 2. 3.05 3. 5.42 4. 4.31 5. 31.91 6. 56 7. 10 8. 10 9. 26 10. 10

11. 165.6 10

12. 186.89 10

13. 214.956 10

14. 169.483 10 15. 3,860,000,000,000 16. 0.000 000 82 17. 425,700,000,000,000

18. 221.72156 10

19. 61.52064 10

20. 332.3814 10

21. 401.729 10 22. 3,240,000

23. 82.603 10 people

63.536 10 square miles =

approximately 73 people/sq mi

24. 231.3 10 25. 0.000 005 4

26. 127.09 10 dollars 292,000,000 people = $24,281 per person

27. 141.3 x 1,000,000 =

141,300,000 = 81.413 10 28. 65 x 1,000,000,000 =

65,000,000,000 = 106.5 10 29. 196 30. 144

31. -256 32. 279,841 33. 4913 34. 10.677 35. 100 36. 108 37. -117 38. -27 39. 81 40. -81 41. 88 42. 1 43. 3 44. 7.48 45. 7

46. 39 is not a real number

47. 3 48. 20 49. 1371 50. 2150 51. 14.42 52. 28.6 53. 29.8 54. 50.5 55. 11.62

56. 86.37 10 57. 1 (Remember that any number

except zero taken to the zero power is one.)

58. 547.75 10 59. 8728.92 60. 66,204.09 61. 97,793.45 62. 9091.34 63. 2970.76 64. 16,325.97 65. 279.22 66. 1060.17 67. 69.81

Unit 1 Calculator Usage and Equation Solving Section 1.2 Rounding off approximate numbers

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Objectives for this section Round off numbers correctly to a given place value Round off result correctly after addition or subtraction Round off units correctly when fractional part is not decimal

Rounding off numbers to a given place value

To understand rounding off, you need to know the place value of each digit in a number. The following example illustrates place value: Example 1. Write the place values of each digit in the following number:

1,593,587.421 067 Answer: (reading the number from left to right)

1 is the millions digit. 5 is the hundred thousands digit. 9 is the ten thousands digit. 3 is the thousands digit. 5 is the hundreds digit. 8 is the tens digit. 7 is the ones or the units digit (or the whole number position). 4 is the tenths digit. 2 is the hundredths digit. 1 is the thousandths digit. 0 is the ten thousandths digit. 6 is the hundred thousandths digit. 7 is the millionths digit.

The normal technique for rounding off numbers to a given digit is to look one digit to the right of the digit that you are rounding. If the digit to the right is 5 or more, increase the rounding digit by one. If the digit to the right is less than 5, leave the rounding digit alone. After rounding, all numbers to the right of the rounding digit are terminated. This procedure is commonly referred to as “rounding up.” When dealing with non-decimal situations such as feet/inches or pounds/ounces, determine if the next number is one half or more of the rounding quantity. In other words, if you wish to round a number to the nearest tenth, you must look at the hundredths digit (and only at the hundredths digit) to decide whether to increase the tenths digit by one or leave it the same.


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Example 2. Round 6.4783259345 to the nearest hundredth.

Answer: 6.48, not 6.480, or 6.4800, or 6.48000, etc. No additional zeros should follow the 8, since 6.480 indicates that the number has been rounded to the nearest thousandth; 6.4800 has been rounded to the nearest ten thousandth; and so on.

Hence, 5.0, 5.00, 5.000, and 5.0000 all represent the same number rounded to different numbers of decimal places. Example 3. Rounding off 4768 to the nearest hundred yields 4800.

Example 4. Rounding off 3.7349999 to the nearest hundredth yields 3.73.

Example 5. Rounding off 45.6736554 to the nearest tenth yields 45.7.

Example 6. Rounding off 137.696 to the nearest whole number yields 138.

Example 7. Rounding off $456.73 to the nearest dollar means the same as rounding off to the nearest whole number and thus yields $457.

Example 8. Rounding off 5

78

inches to the nearest inch yields 8 inches, since 5

8 is

more than 0.5 inches. Example 9. Rounding off $13.36821 to the nearest penny means the same as

rounding off to the nearest hundredth and thus yields $13.37. Example 10. Rounding off 12 yd 2 ft 4 in to the nearest yard yields 13 yd since 2 ft 4

in is more than 0.5 yd. Example 11. Rounding off 8 yd 1 ft 3 in to the nearest yard yields 8 yd since 1 ft 3

in is less than 0.5 yd. Example 12. Rounding off 5.96 to the nearest tenth yields 6.0. The zero after the

decimal must be written to show that the number has been rounded to the nearest tenth, even though 6 is numerically the same as 6.0.

Example 13. Rounding off 384 to the nearest thousand yields 0 since the number is

less than half of a thousand.


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You can check to see if your calculator displays a rounded-off number by dividing out

2

3. If the answer is all 6’s, then your calculator is not displaying a rounded-off

answer. If the last digit is 7, then your calculator is displaying a rounded-off answer. Your calculator will allow you to fix the number of decimal places it displays. Rounding off after Addition or Subtraction

When you use your calculator to do a calculation such as 37.4 + 5.7631 = 43.1631, it is tempting to keep all of the decimal places given by the calculator. You should not do this as the following example shows. Example 14. The number 37.4 could have been as large as 37.449 (which would

have rounded off to 37.4), and the number 5.7631 could have been as large as 5.763149 (which would have rounded off to 5.7631). Adding 37.449 + 5.763149 yields 43.212149, a number which agrees with the previous answer (43.1631) only in the first two digits. Clearly, the answer given by the calculator has too many decimal places considering the precision of the two numbers used to obtain it.

When the problem involves only addition and subtraction of different numbers, the answer should contain only as many decimal places as the fewest decimal places contained in any number used in getting the answer. The numerical value of the last place in a number is called the precision of the number. Example 15. Suppose you add the approximate numbers 4.763 + 12.6 and get the

result 17.363. The first number before it was rounded off could have been as small as 4.7625, and the second number could have been as small as 12.55. Adding these two numbers together gives 17.3125. Notice that this answer agrees with the original answer only as far as the first decimal place, 17.363 and 17.3125. This tells you that you should round the answer to the nearest tenth, which is one decimal place, just as the round off rule for addition and subtraction states.

Example 16. Round off the answer to the calculation:

2.76 + 45.832 + 3.98 453 + 234.223 + 12.34 = 299.13953

The answer should be rounded off to 299.14, since the least precise number used in calculating the answer contained only two decimal places. The precision of the answer is 0.01.


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Example 17. Round off the answer to the calculation 5.438 + 22.3 = 27.738 The answer should be rounded off to 27.7, since 22.3 contains only one decimal place.

Example 18. Round off the answer to the calculation 245.233 4 - 38.715 = 206.5184.

The answer should be rounded off to 206.518, since 38.715 contains only three decimal places.

Special Cases for Rounding Off

This round-off information applies only to approximate numbers such as measure-ments which are only as precise as the measuring instruments. If you know that the numbers you are using are exact numbers, don’t round at all except when a division causes an answer to come out as a long decimal. Also, money is generally rounded to the nearest penny or to the nearest dollar. If you are working in a situation where a certain round-off technique is mandated by company policy, (federal tax rules, etc.), then obviously you must follow whatever rules the job requires. For instance, there may be a policy to round all calculations to two decimals regardless of how they were obtained. Rounding When a Method is Specified or Numbers are Exact

Example 19. An Internal Revenue Service instruction on a tax form instructs you to round off all fractions obtained to 4 decimal places. How should you round off the result of dividing $28,264 by $42,321?

Answer: Dividing 28,264 by 42,321 gives 0.667848113. Rounding this number to

4 decimal places gives 0.6678, because the number following the fourth decimal place is less than 5.

Example 20. If you buy 14 items which cost $5.76 each, how much will you spend?

In this problem, both the number, 14, and the price, $5.76, are exact numbers which have not been rounded off.

Answer: 14 5.76 = $80.64, which is the exact answer and should not be rounded at all.

Example 21. If there are 22 rooms and each one holds 44 persons, what is the total

number of persons in all of the rooms? Answer: 22 44 = 968 persons, which is an exact answer and should not be

rounded.


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What if you were asked to round by 5’s? This would mean that the number 7 would round to 5 and the number 8 would round to 10. Example 22. If you have 297 objects in one container and 351 objects in another

container, estimate the number of total objects rounded to the nearest 5 objects.

Answer: 297 + 351 = 648. The rounded answer would be 650 objects.

You may be asked to round measurements to the nearest 1

4 inch. For the respective

diagrams, the measurements on the ruler are as indicated. 0.25 inch 0.5 inch 0.75 inch

Example 23: For the following diagram, round the measurement to the nearest 1

4

inch.

Answer: The length of the line segment is 3

38

inches. When this measurement is

rounded to the nearest 1

4 inch, it is rounded up to

43

8 inches which

equals 1

32

inches, or 3.5 inches as a decimal.


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ALWAYS INCLUDE UNITS IN YOUR ANSWER IF UNITS ARE PROVIDED IN THE PROBLEM! Round off the following as directed:

1. 6.0592 to the nearest hundredth

2. 58.4751 to two decimal

places 3. 285,928 to the nearest

thousand 4. 2491.683 to the nearest

whole number 5. 244.999 to the nearest ten

6. 799.9996 to the nearest thousandth

7. 0.003 to the nearest

hundredth 8. 0.207 to the nearest whole

number (sometimes called the nearest unit or the nearest digit)

9. 0.149999 to the nearest tenth 10. $265.89 to the nearest dollar

11. $35.48 to the nearest dollar

12. $14.3628 to the nearest cent

13. $3.4578 to the nearest cent

For the following problems note that:

1 ft = 12 in 1 lb = 16 oz 1 gal = 4 qt

14. 9 ft 2 in to the nearest foot

15. 8 ft 3

54

in to the nearest inch


17. 3

78


18. 1

102



20. 7 ft 7

58

in to the nearest foot

21. 6 lb 4 oz to the nearest pound




25. 5 gal 1 qt to the nearest gallon




35

Round the following according to the rules in your text concerning measured numbers:

28. 0.344 5 2.57 1.486

29. 0.1 0.10 0.100 0.100 0

30. 0.005 0.007 1.630 0.200

31. 58.0 3.863

32. 0.00793 0.002

33. There are 47 jars and each contains 138 marbles. How many marbles are there altogether?

34. Thirty seven persons each have

identical yearly salaries of $39,160.85. Find the total of all of the salaries earned by the persons.

35. Round 235,658 to the nearest

hundred. 36. Round 235,658 to the nearest

thousand.

37. Round 4.0398 to the nearest

tenth. 38. Round 7.026831 to the nearest

hundredth. 39. Round 0.004672 1 to four

decimal places. 40. Round 488 to the nearest

thousand. 41. Round 7 ounces to the nearest

pound. 42. Round 12 fluid ounces to the

nearest pint. 43. Round 5 inches to the nearest

foot. 44. Round 2 feet to the nearest

yard. 45. Round 1 pint to the nearest

quart.


36

46. Round 2 quarts to the nearest gallon.

47. Round 3000 feet to the nearest

mile.

48. Round 11

32 inch to the nearest

inch.

49. Round 19

32 inch to the nearest

inch. 50. Round $2.43624 to the nearest

cent. 51. Round 2576 to the nearest 5.

52. Round 983 to the nearest 5.

53. Find the actual measurement of the following line segment


4 inch.

54. Find the actual measurement of

the following line segment


4 inch.

Unit 1 Calculator Usage and Equation Solving Section 1.2 Answers to Selected Problems

37

1. 6.06

2. 58.48

3. 286,000

4. 2492

5. 240

6. 800.000 (all of the zeros must be present)

7. 0.00

8. 0

9. 0.1

10. $266

11. $35

12. $14.36

13. $3.46

14. 9 ft

15. 8 ft 6 in

16. 14 ft

17. 7 in

18. 11 in

19. 5 ft

20. 7 ft

21. 6 lb

22. 12 lb

23. 9 lb

24. 24 lb

25. 5 gal

26. 7 gal

27. 9 gal

28. 4.40

29. 0.4

30. 1.842

31. 54.1

32. 0.006

33. 47 138 6,486

(both numbers are exact)

34. 37 $39,160.85

$1,448,951.45

35. 235,700

36. 236,000

37. 4.0

38. 7.03

39. 0.0047

40. 0

41. 0 lb

42. 1 pt

43. 0 ft

44. 1 yd

45. 1 qt

46. 1 gal

47. 1 mi

48. 0 in

49. 1 in

50. $2.44

51. 2,575

52. 985

53. 1

14

inch

54. 3 inches

Unit 1 Calculator Usage and Equation Solving Section 1.3 Variables and Formulas

39

Objectives for this section Write verbal statements in symbolic form Evaluate symbolic expressions Combine like terms correctly Use the distributive property

Writing English language expressions using variables

A variable is just a letter used in place of a number. For instance, “triple a number” could be written as “3x,” meaning “3 times x.”

The expression “4 more than a number” could be written “n+4.”

“A number subtracted from 9” could be written “9 – y.”

“Five less than a number” could be written “n – 5.”

“An integer added to the next integer” could be written “x + x + 1.”

“Three more than double a number” could be written “2z+3.”

“Six divided by a number” could be written “6

x.”

“A number divided by six” could be written “6

x”

“The sum of a number and 17” could be written “y + 17.”

Example 1. Write an expression for “five less than four times a number.”

Answer: Let z be the number. 4z – 5

Since you will be working with variables as you set up and solve equations throughout the course, you need to know some facts concerning variables. An upper case letter such as “B” is considered to be a completely different variable from the same lower case letter “b.” In other words, B and b could represent two different numbers. You should treat them as if they were different letters. Likewise,

a letter with a subscript such as “ 1y ” is considered to be a different variable from the

same letter with a different subscript such as “ 2y .” A subscript is simply a small

number written at the bottom of the letter. As we solve equations later, variables will be called unknowns.


40

Evaluating Expressions involving Variables

The word “evaluate” means to replace the variables (letters) by whatever numbers are given for those variables and then to calculate the resulting number. Example 2. Evaluate the expression 4x2y + 3y2 for x = 5 and y = 6.

Answer: 2 24(5 )(6) 3(6) 4(25)(6) 3(36) 600 108 = 708

Example 3. If T = 4(a + b – 3), find T when a = 10 and b = 13.

Answer: T = 4(10 + 13 – 3) = 4(20) = 80. Interpret “find” as “evaluate.”

Example 4. The target heart rate for a man, where A is the age of the man, is given by the formula, H = 0.75(220 – A). Find the target heart rate for an 18-year-old man.

Answer: H = 0.75(220 – 18) = 0.75 202 = 151.5 or 152 rounded to nearest whole number.

Example 5. What is the target heart rate for a 48 year old?

Answer: H = 129

Accounting may be considered the global language of business. Every piece of data collected in any business accounting system is reported to someone somewhere in the world. Attempting to understand such a data collecting and reporting system may sound overwhelming, but the entire accounting discipline is based on one equation: Assets = Liabilities + Equity. Assets are the resources we convert to products and services that we sell to consumers. Liabilities are what we owe on the Assets. Equity is what we own of the Assets. Example 6. What are the Assets for a company whose Liabilities are $27,215 and

whose Equity is $54,905? Answer: Assets = Liabilities + Equity. Assets = $27,215 + $54,905 = $82,120


41

Adding and subtracting terms containing variables

An expression consisting of some numbers and variables multiplied together is called a term. When terms are added together, only those terms containing exactly the same variable to the same power may be added together. Such terms are called “like terms.” The variable parts of the expressions which are added or subtracted consist of exactly the same letters, exponents, subscripts, and are the same case. The number in front of the variable is called the numerical coefficient of the variable. The expression 15x is a term. The numerical coefficient of x is 15. Often, the numerical coefficient is just called the coefficient. The following examples are easy to understand if you remember that an expression such as 4x really just means x + x + x + x. Example 7. 5x + 4x = 9x

Example 8. x + 0.7x = 1.7x (Remember that x = 1x.)

Example 9. 5a + 3A + 2a + 9A = 7a + 12A (cannot be combined further because upper case and lower case letters are not “like terms”)

Example 10. 2 2 2 1 24 6 5 2x y x x y 2 2 19x +5y +2x (cannot be combined further

because letters with different subscripts are not “like terms”)

Distributive Property of Numbers

There is an algebra rule called the distributive property which is used when multiplying terms inside parentheses by something on the outside of the parentheses. To multiply 3(2x + 5), multiply the number on the outside times each of the terms on the inside of the parentheses to get the result 6x + 15. This property is demonstrated by the following numerical examples:

Example 11. 5(7 + 3) = 5(10) = 50

5(7) + 5(3) = 35 + 15 = 50

Example 12. Multiply 6(5x + 4y + 2)

Answer: 30x + 24y + 12

A(B + C) = AB + AC


42

1. Express using a variable: Five more than twice a number.

2. Express using a variable: The product of two consecutive integers.

3. Express using a variable: The sum of two consecutive integers.

4. Express using a variable: The sum of two consecutive odd integers.

5. Express using a variable: Eight less than the square of a number.

6. Express using a variable: Twice a number plus three times the square of the same number.

7. “Perimeter” means the distance around a figure. Using the letters shown,

write an expression for the perimeter of the figure shown.

8. Evaluate 33xy 4x -5 when x 2 and y 3 .

9. Evaluate a b

ab when a = 4 and b = 9.

10. Evaluate 4a 5b when a = 2.57 and b = 3.28.

11. Use the formula for the target heart rate for a female H = 0.65(220 – A), where A is the age in years, to find the target heart rate for a 30-year-old female.

12. A formula for recommended weight according to height is

11 (h 40)

W2

,

where h is the height in inches, and W is the weight in pounds. Find the recommended weight for a person who is 5 ft 6 in tall.

13. A formula for calculating the bowling handicap for a bowler is

h = 160 - 0.8a. The bowler’s handicap is represented by h and the average score is represented by a. Find the handicap for a bowler whose average score is 140. Round the result to the nearest whole number.


43

14. Using the formula given in problem #13, find the handicap for a bowler who scored 158, 149, 139, 160, 141, and 150 in the last six games. (Compute the average first.)

15. The systolic blood pressure is the force with which blood is pumped from the

heart. One formula for calculating what this pressure should be is

a

P 1002

, where P is the systolic blood pressure, and a is the age of the

patient in years. Find the systolic blood pressure for a 40-year-old patient. 16. Given the Accounting formula Assets = Liabilities + Equity, find the Assets of a

company whose Liabilities are $153,200 and whose Equity is $510,480. 17. Combine 5x + 7y + 2x + 9y.

18. Combine 3.4x + 6.7y – 2.1y + 5.3x.

19. Combine 1 13x 9x 7x 4x .

20. Combine 6X + 4X + 2x – 8x.

21. Multiply 5(2x + 5).

22. Multiply 3(7 – 4 x).

23. Multiply 4(5x – 7).

24. Multiply and combine: 3(5 – 4x) + 2(9x – 1).

25. Multiply and combine: 8(3x + 4y) – 4(y – x).

26. What is the coefficient of x in the expression -7x?

27. Combine x – 0.3x.

28. Combine p + 0.7p.

29. Combine a – 0.2a.

30. Combine y + 1.3y.


44

31. Combine d + 2.4d.

32. Combine n – 0.9n.

33. Combine 3x + 4x + 2 + 6x + 8.

34. Combine 8p + 16 - 2p – 5.

35. Express using a variable: "The sum of a number and another number which is ten more than the original number."

36. Express using a variable: "A number added to 3.5 times the same number."

37. Express using a variable: "A number increased by 0.7 times the number."

38. Express using a variable: "A number decreased by 0.3 times the number."

39. Express using a variable: "A number decreased by seven tenths of the number."

Round the answers to the following problems to one decimal place.

40. Evaluate (a + b)c + d when a = 2.85, b = 1.48, c = 3.47, d = 7.21.

41. Evaluate (a + b)(c + d) when a = 2.85, b = 1.48, c = 3.47, d = 7.21.

42. Evaluate a + b(c + d) when a = 2.85, b = 1.48, c = 3.47, d = 7.21.

43. Evaluate a + bc + d when a = 2.85, b = 1.48, c = 3.47, d = 7.21.

44. Evaluate abc + d when a = 2.85, b = 1.48, c = 3.47, d = 7.21.

45. Evaluate abcd when a = 2.85, b = 1.48, c = 3.47, d = 7.21.

46. Evaluate ab + cd when a = 2.85, b = 1.48, c = 3.47, d = 7.21.


45

1. 2x + 5

2. (x)(x + 1)

3. x + x + 1

4. x + x + 2

5. 2x 8

6. 22x 3x

7. h + b1 + s + b2

8. 45

9. 13 1

26 6

10. 5.17, rounded to two decimal places

11. 124

12. 143 pounds

13. 48

14. 40

15. 120

16. $663,680

17. 5x + 2x + 17y +9y = 7x +16y

18. 8.7x +4.6y

19. 110x 5x

20. 10X – 6x

21. 5(2x) + 5(5) = 10x + 25

22. 21 - 12x

23. 20x – 28

24. 15 – 12x + 18x – 2 = 13 + 6x

25. 24x + 32y –4y + 4x = 28x + 28y

26. -7 (the coefficient includes the minus sign if there is one present)

27. 0.7x

28. 1.7p

29. 0.8a

30. 2.3y

31. 3.4d

32. 0.1n

33. 13x + 10

34. 6p + 11

35. x + 10 + x

36. n + 3.5n

37. p + 0.7p

38. x – 0.3x

39. n - 0.7n

40. 22.2

41. 46.2

42. 18.7

43. 15.2

44. 21.8

45. 105.5

46. 29.2

Unit 1 Calculator Usage and Equation Solving Section 1.4 Equations

47

Objectives for this section Solve linear equations Solve stated problems by using linear equations

An equation is an expression containing an equal symbol (=).

Examples are: (5)(20) = 100, or 2 + 3 = 5. However, these examples do not represent the type of equation in which we are interested. The type of equation that we will use in applications will contain an unknown such as 5x = 100. (Remember that 5x means 5 times x.) Solving Equations

A solution of an equation containing a variable or unknown is a value of the unknown which makes the equation true. Obviously x = 20 is a solution of the equation 5x = 100, since 5(20) = 100.

We will be working with equations such as 4x = 24, and will be finding what value of x makes the equation true. Finding this value of x, which is 6 in this case, is called solving the equation. Solving an equation means performing a series of operations which isolate the unknown or make the unknown appear by itself. All of the equations we will be solving are what are known as linear equations. That is, the unknown is to the first power and no other power. Such equations have only

one answer. Equations such as 2x 25 , which are not linear, may have more than

one solution. In this case, 2x 25 has solutions 5 and -5. The same quantity may be added, subtracted, multiplied or divided on both sides of the equal symbol in an equation without changing the solution to the equation. You cannot multiply or divide both sides of an equation by zero or by any letter which might be equal to zero. Example 1. Solve the equation x + 5 = 20.

Answer: x + 5 – 5 = 20 – 5 Subtract 5 from both sides x + 0 = 15 Combine x = 15 This is the solution.

The reason that we subtracted 5 from both sides was because 5 was added to the unknown originally, and subtraction is the opposite operation from addition. Thus, subtracting 5 makes the unknown appear by itself on the left side of the equation.


48

Example 2. The equation x - 6 = 19 may be changed to the equation x = 25 by adding 6 to both sides. The solution of this equation is x = 25. The reason for adding 6 to both sides is that in the original equation 6 is subtracted from the unknown x, and addition is the opposite operation from subtraction. Note that x - 6 + 6 = x (subtracting 6 and adding 6 equals 0), and x plus zero equals x.

Example 3. Solve the equation x + 6 = 15.

Answer: Since 6 is added to x, you need to subtract 6 from both sides in order to find x.

This gives the result x + 6 – 6 = 15 – 6.

Simplifying yields the solution of the equation, x = 9.

Example 4. Solve the equation x – 2 = 8.

Answer: Since 2 is subtracted from x, you need to add 2 to both sides in order to find x.

This gives the result x – 2 + 2 = 8 + 2.

Simplifying yields the solution, x = 10.

Example 5. Solve the equation 3x = 24.

Answer: 3x 24

3 3 Because x is multiplied by 3, and division is the

x = 8 opposite operation from multiplication, divide both sides by 3 in order to isolate x. (Note that

3x 3 = x, because 3 divided by 3 = 1)


49

Example 6. Solve the equation x

54 .

Answer: Because x is divided by 4, and multiplication is the opposite of division, multiply both sides by 4 in order to isolate x.

This gives the result

x4 4 5

4.

Simplifying yields x = 20.

Example 7. The equation x

58

may be changed to the equation x = 40 by

multiplying both sides by 8. Note: x

8 x8

, because 8 divided by 8

equals 1. Thus, the solution to the equation is x = 40.

Example 8. Often an equation will require more than one operation to solve.

Solve 5x + 7 = 27

Answer: 5x + 7 – 7 = 27 – 7 Subtract 7 from both sides 5x = 20 Combine

5x 20

5 5 Divide both sides by 5

x = 4

Sometimes the unknown will appear on both sides of the equation. In such a case, the equation must be changed so that the unknown appears on only one side. Example 9. Solve 8x + 2 = 17 + 5x.

Answer: 8x – 5x + 2 = 17 + 5x – 5x Subtract 5x from both sides 3x + 2 = 17 Combine 3x + 2 – 2 = 17 – 2 Subtract 2 from both sides 3x = 15 Combine x = 5 Divide both sides by 3


50

Example 10. Solve the equation 5x + 2 = 3x + 12.

Answer: 2x + 2 = 12 Subtract 3x from both sides so that the unknown appears only on the left side. 2x = 10 Subtract 2 from both sides x = 5 Divide both sides by 2

Example 11. Other times the equation may contain parentheses which must often be multiplied out using the distributive property before the equation can be solved.

Solve 2(3x + 5) – 7 = 63.

Answer: 6x + 10 – 7 = 63 Multiply using distributive property 6x + 3 = 63 Combine 6x + 3 – 3 = 63 – 3 Subtract 3 from both sides 6x = 60 Combine

6x 60

6 6 Divide both sides by 6

x = 10

Example 12. Solve the equation 3(x + 4) = x + 20.

Answer: 3x + 12 = x + 20 Multiply using the distributive property 2x + 12 = 20 Subtract x from both sides 2x = 8 Subtract 12 from both sides x = 4 Divide both sides by 2

If there is no number in front of the unknown, then it is understood that there is a one (1) in front. Hence, x may be read as 1x. Some examples where this property proves useful follow: Example 13. x + 5x = 6x (Since 1x + 5x = 6x)

Example 14. x + 0.3x = 1.3x (Since 1x + 0.3x = 1.3x)

Example 15. x – 0.6x = 0.4x (Since 1x – 0.6x = 0.4x)

Example 16. Solve x + 0.5x = 300.

Answer: 1.5x = 300 Since 1x + 0.5x = 1.5x

x = 200 Divide both sides by 1.5


51

Example 17. Solve x = 0.8x + 200.

Answer: x – 0.8x = 200 Subtract 0.8x from both sides 0.2x = 200 1x – 0.8x = 0.2x x = 1,000 Divide both sides by 0.2

Do not expect equations to contain only whole numbers. Since you should be familiar with your calculator, this should not pose a problem, provided you know the procedure for solving equations. See the next example. Example 18. Solve 1.4(2x – 3) + 5.9 = 0.6x + 13.8.

Answer 2.8x – 4 .2 + 5.9 = 0.6x + 13.8 Multiply using the distributive property 2.8x + 1.7 = 0.6x + 13.8 Subtract 4.2 from 5.9 2.2x + 1.7 = 13.8 Subtract 0.6x from both sides 2.2x = 12.1 Subtract 1.7 from both sides x = 5.5 Divide both sides by 2.2

When you solve an equation, you need not tell what you do in each step. You should, however, perform only one operation in each step and show the result of each step. Remember: Do not round until you reach the final answer in the equation. In other words, you should only round the final answer not any intermediate answers. Although textbooks often prefer that the unknown appear on the left side in the solution to an equation, it really makes no difference on which side of the = (equal) symbol the unknown appears. The variable should appear only once with a coefficient of one. In other words, the answer 7 = x is just as good as the answer x = 7.

Example 19. Solve the equation

x 100

2.450

.

Answer:

x 10050 2.4 50

50 Multiply both sides by 50

-120 = x – 100 -120 + 100 = x – 100 + 100 Add 100 to both sides -20 = x


52

Example 20. Solve the equation

x 3

x 15

.

Answer: x + 3 = 5(x + 1) Multiply both sides by 5 x + 3 = 5x + 5 Note that 5 must be multiplied

times both the x and the 1 3 = 4x + 5 Subtract x from both sides -2 = 4x Subtract 5 from both sides -0.5 = x Divide both sides by 4

Example 21. Let’s assume that you purchase a landscaping business that you intend to operate part-time in addition to your full-time job. The business contains the following Assets and Liabilities:

Landscaping Value

Assets $24,555.00

Liabilities $10,320.00

Using the Accounting formula, Assets = Liabilities + Equity, what will be your equity?

Answer: $24,555.00 = $10,320.00 + Equity

$24,555.00 - $10,320.00 = Equity $14,235.00 = Equity

The Income Statement of a business subtracts Expenses from Revenue. Revenue is the amount a business earns from services, and expenses are the amounts paid to provide those services. When Revenue is greater than Expense, the company has a Net Profit. When Expense is greater than Revenue, the company has a Net Loss. Example 22. If your company has Revenue of $28,566 and Expenses of $11,899 the

first year, what is your net loss or profit? Answer: Net profit is $28,566 - $11,899 = $16,667.

Creating equations

Sometimes, a problem will not be stated in the form of an equation, and you must create the equation yourself. Occasionally, creating the equation will involve some special knowledge not contained in the problem, such as the formula for the perimeter of a square, the meaning of a word, some financial formula, etc. Some examples of problems where you must set up an equation follow:


53

Example 23. Suppose you own a small business that had a profit of $950 in the month of March. How much did the company’s Equity increase?

Answer: Equity is increased if a company has a profit and is decreased if a

company has a loss. Therefore, because the company had a profit of $950 in the month of March, the Equity increased by $950.

Example 23. Suppose the perimeter of a square is 80 feet. Create an equation to

find the length, x, of one side of the square. Answer: Since the perimeter of a square is four times the length of one of its

sides, the equation is 4x = 80. Example 24. Seventeen less than twice a number is 53. Create an equation to find

the number. Answer: Let x = the number. 2x – 17 = 53 Example 25. George has $13.50 worth of quarters. Create an equation to find how

many quarters he has. Answer: Let x = the number of quarters

0.25x = 13.50 since each quarter is worth $0.25.

Note that both sides of the equation must be in the same units - dollars in this case. Another correct equation, using cents as units, would be 25x = 1350.

Example 26. The sum of two consecutive integers is 143. Create an equation to find

the two integers. Answer: To solve this problem, you must realize that the expression,

“consecutive integers,” means a pair of numbers like 21 and 22 or 44 and 45. If we decide to call the first integer “n,” then the next integer must be one more than “n” or “n + 1.” Since “sum” means to add, the equation is n + n + 1 = 143.


54

Example 27. We wish to cut a 5 meter board into two pieces so that the longer piece is 1.6 meters longer than the shorter piece. Create an equation to find the lengths of the two pieces.

Answer: Call the length of the shorter piece x, and the length of the longer

piece x + 1.6. The equation then becomes x + x + 1.6 = 5 Example 28. A number added to 0.07 times the same number gives a result of 856.

Create an equation to find the number. Answer: In words, the equation is “the number plus 0.07 times the number

equals 856.” The equation is x + 0.07x = 856.

General Procedures for Solving Equations:

1. Multiply out parentheses first.

2. If the unknown appears on the bottom, multiply both sides by the unknown so that it appears on the top before you try to solve the equation.

3. If there are any other fractions present, multiply both sides by the common

denominator for all fractions. 4. Whenever the unknown appears on both sides of the equal symbol in the

original equation, add or subtract on both sides so that the unknown appears on only one side.

5. To eliminate a number added to the term containing the unknown, subtract

the number from both sides. 6. To eliminate a number subtracted from the term containing the unknown,

add the number to both sides. 7. To eliminate a number multiplied times the unknown, divide by the number

on both sides. 8. To eliminate a number divided into the unknown, multiply by the number

on both sides.


55

Solving Word Applications

1. Read the problem, several times if necessary, until you understand what is given and what is to be found.

2. Assign a variable to represent the unknown quantity. Use pictures as

necessary. Write down what the variable represents. 3. Write the equation using the variable from step 2.

4. Solve the equation for the variable.

5. State the answer to the problem including any units that are needed. This may include additional answers other than the one found in step 4. Is the answer reasonable?

6. Check the answer(s) in the words of the original problem.


56

Solve for the variable and round to two decimal places if the answer is a non-terminating decimal. 1. 5x = 60

2. 4x = 100

3. 3y = 12

4. 0.2p = 5

5. x + 9 = 200

6. x + 6 = 13

7. y + 1.3 = 4.8

8. a + 2.5 = 7.6

9. c + 8 = 5

10. x – 77 = 105

11. x – 5 = 60

12. d - 1.7 = 3.2

13. u – 4.6 = 3.7

14. x

506

15. x

103

16. v

45

17. x + 5.3 = 19.4

18. 2.6x = 143.4

19. x – 2.683 = 4.729

20. x

5.56353.7456

21. 3x + 8 = 38

22. 3.78x – 2.44 = 6.57

23. 19x + 35 = 4x + 110

24. 6x + 80 = 10x – 20

25. 2.36(x - 1.5) = 0.88x + 9.22

26. 3(x + 2) = x + 40

27. x + 0.8x = 720

28. x + 0.05x = 71.4

29. x – 0.2x = 24

30. 300 + 0.4x = x

31.

x 1000

0.8200

32.

x 400

1.2100

33. x + 2x + x + x + 20 = 180

34. 3x + 20 + x + x - 40 = 180

35. The sum of two consecutive integers is 517. Find them.

36. The perimeter of a square is 424

feet. Find the length of one side of the square.


57

37. Edna has $2.85 worth of nickels.

How many nickels does she have? 38. Thirteen less than five times a

number is 492. Find the number.

39. The sum of two consecutive odd

integers is 244. Find the two integers.

40. In 8 years Bill will be 35 years

old. How old is he now? 41. We wish to cut a 47 inch piece of

string into two pieces so that one piece is 17 inches longer than the other. How long must each piece be?

42. Gigi has $58.50 worth of half

dollars. How many does she have?

43. Matilda is 6 years older than

Bob. The sum of their ages is 28. Find both of their ages.

44. Sally has twice as much money

as Ann has. Together they have $51. How much money does each have?

45. Carrie has half as much money as

Dwyane has. Together they have $123. How much money does each have?

46. Beth wishes to cut a ribbon into

two pieces so that one piece is three times as long as the other piece. If the ribbon is 112 cm long, how long will each piece be?

47. In addition to her salary Sarah

receives an expense account which is equal to one fourth of her monthly salary. If she receives a total of $7100 per month, how much is her salary?

48. The value of Jack’s house is

$13,000 less than twice the value of Bill’s house. If the value of Jack’s house is $245,000, find the value of Bill’s house.

49. Using the equation Assets =

Liabilities + Equity, find the missing values in the following table.

In Millions of Dollars

Assets Liabilities Equity

AT&T 277,787 ?? 91,482

McDonald's ?? 20,617 16,009

Target 44,553 28,322 ??

50. Your bookkeeper tells you,

“Congratulations, your ownership increased by $9457 this year.” If your Assets were $24,555 and your liabilities were $10,320 at the beginning of the year, what is your Equity at the end of the year?


58

51. This year your part-time landscaping business has Assets totaling $24,555, Liabilities totaling $10,320, and Equity totaling $14,235. If next year your Assets increase by $3200 and your Equity decreases by $1890, what will be your total Liabilities next year?

52. If your small business has total

revenue of $72,180 and total expenses of $27,175, did the business earn a profit or a loss?

53. If a company’s Equity at the end

of 2013 was $11,800, and the company had a total Profit for 2014 of $45,005, what is the company’s Equity at the end of 2014?

54. If a company’s Equity at the end

of 2013 was ($11,800), negative Equity, and the company had a total Profit for 2014 of $45,005, what is the company’s Equity at the end of 2014?

55. Solve

a) 5x + 2 = 22

b) 7 – 4x = 2x – 35

c) 8x + 14 = 3x + 54

d) x + 1.4x = 560

e) p – 0.35p = 260

f) 600 + 0.7 p = p

g) a + a + 30 + 2a = 110

h) 3(2x – 5) + 4x = 85

i) 700 5

x 8

j) 7 84

15 x


59

1. x = 12 Divide by 5

2. x = 25 Divide by 4

3. y = 4

4. p = 25

5. x = 191 Subtract 9

6. x = 7 Subtract 6

7. y = 3.5

8. a = 5.1

9. c = -3

10. x = 182 Add 77

11. x = 65 Add 5

12. d = 4.9

13. u = 8.3

14. x = 300 Multiply by 6

15. x = 30 Multiply by 3

16. v = -20

17. x = 14.1 Subtract 5.3

18. x = 55.15 Divide by 2.6 and round

19. x = 7.41 Add 2.683

20. x = 20.84

21. x = 10 Subtract 8 and divide by 3

22. x = 2.38 Add 2.44, divide by 3.78, and round

23. 15x + 35 = 110 Subtract 4x 15x = 75 Subtract 35 x = 5 Divide by 15

24. 25 = x Subtract 6x, add 20, divide by 4

25. 2.36x – 3.54 = 0.88x + 9.22 Multiply out parentheses

1.48x – 3 .54 = 9.22 Subtract 0.88x

1.48x = 12.76 Add 3.54

x = 8.62 Divide by 1.48 and round.

26. x = 17 Multiply out parentheses, subtract x, subtract 6, divide by 2

27. 400

28. 68

29. 30

30. 500

31. 1160

32. 520

33. 32

34. 40


60

35. Let the first integer be x and the second integer be x + 1. Then the equation is x + x + 1 = 517. Solving yields 258 and 259 as the two answers.

36. The perimeter of a square is

four times the length of a side. Hence, the equation is 4x = 424, and the solution is 106 feet.

37. Since a nickel is worth 0.05

dollars, the equation is 0.05x = 2.85, and the solution is 57 nickels.

38. 5x – 13 = 492

x = 101

39. If the integers are odd, then the larger one must be 2 more than the smaller one. Hence, the equation is: x + x + 2 = 244. The two integers are 121 and 123.

40. x + 8 = 35 x = 27 years old

41. x + (x + 17) = 47 x = 15 inches x + 17 = 32 inches

42. 0.50x = 58.50 x = 117 half dollars

43. b + (b + 6) = 28 b = 11 years (Bob) b + 6 = 17 years (Matilda)

44. a + 2a = 51 a = $17 (Ann) 2a = $34 (Sally)

45. d + 0.5d = 123 d = $82 (Dwyane) 0.5d = $41 (Carrie)

46. x + 3x = 112 x = 28 cm 3x = 84 cm

47. S + 0.25S = 7100 1.25S = 7100 S = $5680

48. 2B – 13,000 = 245,000 2B = 258,000 B = $129,000

49. Liabilities for AT&T $186,305 million Assets for McDonald’s $36,626 million Equity for Target $16,231 million

50. $23,692

51. $15,410

52. $72,180 - $27,175 = $45,005 Revenue is greater than expenses. Therefore, the business earned a profit.

53. $11,800 + $45,005 = $56,805

54. ($11,800) + $45,005 = $33,205

Unit 1 Calculator Usage and Equation Solving Section 1.5 Introduction to percent

61

Objectives for this section Change percents to ordinary numbers Change ordinary numbers to percents Solve the three basic percent problems (find a% of b, a is what % of b, a is b% of what number)

Percent

A basic concept necessary for understanding many everyday uses of mathematics is the idea of percent. The word “percent” comes from the Latin “per centum” which

means per hundred. The symbol used for percent is %, and it means 1

100 or 0.01. For

instance, 80% means 1

80100

or 80

100 or

8

10 or

4

5 or 0.8, depending upon which way

you choose to write the fraction. The number 6 percent means 6 parts out of 100 parts or the ratio 6 to 100. When you “take 30% of a number,” what you really do is

multiply the number by 30 x 0.01 or 0.3 or 3

10. Generally, for such simple

operations, you can just use the % key on your scientific calculator, since it automatically changes 30% to the decimal fraction 0.3. Hence, taking 30% of 80 would involve multiplying 30 x 0.01 x 80, which would give the answer 24. In general, the word “of” means to multiply. One way to handle problems involving percent is to replace the percent symbol

(%) with 0.01 or 1

100 whenever it occurs in a calculation or in an equation. One

method for changing from decimal to percent is to move the decimal point two places to the right and add the percent sign. To change from a percent to a decimal, move the decimal point two places to the left and remove the percent sign. Example 1. Change 65% from a percent to a fraction.

Answer:

1 6565% 65

100 100

13

20


62

Example 2. Change 3

8 from a fraction to a percent.

Answer: First, change the fraction to a decimal. 3

0.3758

. Next, move the

decimal two places to the right and add the percent sign. Thus,

3

0.3758

37.5% .

Example 3. Change 0.07% from a percent to a fraction.

Answer:

1 7 10.07% (0.07)

100 100 100

7

10,000.

Example 4. Doubling a number is the same as multiplying by what percent?

Answer: Doubling a number means multiplying it by 2. Convert 2 to a percent by moving the decimal two places to the right and add the percent sign. Thus, 2 = 200%. This means that doubling is the same as multiplying by 200%.

Example 5. Taking half of a number is the same as multiplying the number by what

percent?

Answer: Convert 1

2to a decimal, 0.5. Move the decimal two places to the right

and add the percent sign. Thus, taking half is the same as multiplying by 50%.

Example 6. Multiplying a number by three fourths is the same as multiplying it by

what percent?

Answer: Convert 3

4 to a decimal, 0.75. Move the decimal two places to the right

and add the percent sign. Thus, three fourths is 75%.


63

Example 7. Multiplying a number by 600% is the same as multiplying the number by what?

Answer: To change 600% to a number, you move the decimal two places to the

left. Thus, 600% = 6. Because % can be calculated by 0.01, you could also do the following calculation 600(.01) = 6.

Example 8. Change 1

%3

to a common fraction.

Answer: If you wish to change a number to a fraction, it may be easier to use

1

1%100

. Thus,

1 1 1%

3 3 100

1

300.

Note that if we were to change this percent to a decimal, we would get 0.003333333333... There would be no rule for rounding off the decimal. The rule may well depend upon where you are using the percent. For instance, the rules for percents when computing income tax may instruct you to round off the decimal form of the percent at the fourth or fifth decimal place. So, in this instance we would round off the answer to 0.0033 or 0.00333.

Example 9: Change

1 1 1%

17 17 100

1

1700.

Elementary Problems involving Percent

The three most basic types of percent problems are:

1. Find some percent of a number. 2. Find what percent one number is of another number. 3. If a known number is a given percent of some unknown number, find

the unknown number.


64

Example 10. Find 150% of 500.

Answer: Because “of” means to multiply, (150%) (500) = (1.5)(500) = 750.

Example 11. Find 1

% of 40005

.

Answer:

1% 4000 (0.2%)(4000) (0.002)(4000)

58 . Or,

1 1 1% 4000 (4000)

5 5 1008 . Or,

1 1% 4000 (0.01)(4000)

5 58. Notice that there are several

different ways to work the same problem. All of these methods of converting percents are equivalent.

Example 12. Find 40% of 90.

Answer: (0.4)(90) = 36. Or, (40)(0.01)(90) = 36.

Example 13. What percent of 80 is 16?

Answer: Write the equation:

Let x = the unknown %

(x)(80) = 16 80x = 16 x = 0.2 x = 20% In other words, 16 is 20% of 80.

Example 14. 60 is 15% of what number?

Answer: Write the equation 60 = 15% of x, remembering that “is” means “equals” and x is the number.

60 = (0.15)(x) Change 15% to 0.15

60

x0.15

400 = x


65

Applications

A person who is self-employed must pay 15.3% of net income for social security and medicare taxes in addition to normal income taxes. Example 15. Find the amount of social security and Medicare taxes paid by a self-

employed person whose net income is $20,000. Answer: Multiply 15.3% times $20,000 to get $3060.

Example 16. Suppose the person in the previous example also pays 15% in federal income tax and 5% in state income tax. What will be the total amount of tax paid, including social security, Medicare, federal, and state taxes?

Answer: federal tax (15%)(20,000) = (0.15)(20,000) = $3000

state tax (5%)(20,000) = (0.05)(20,000) = $1000 social security and medicare from previous example = $3060

Add all three taxes $3060 + $3000 + $1000 = $7060.

When a lender such as a bank lends money to purchase a house, it often charges what are called loan origination fees or points. A charge of 1 point is one percent of the amount of the loan. For instance, if the lender is lending $80,000, a fee of 2 points would equal 2% of $80,000 or (0.02)(80,000) or $1600. This is money collected by the lender in addition to the normal interest charges.

Example 17. Calculate the amount charged by a lender on a $90,000 home loan if the charge is 1.5 points.

Answer: Multiply 1.5% times 90,000.

(0.015)(90,000) = $1350


66

Example 18. At the end of the first year of operations of our part-time landscaping business, the Balance Sheet reports the following Assets, Liabilities and Equity.

Acct. # Account Name Blank

100 Cash $6355.00

110 Accounts Receivable 150.00

140 Supplies 50.00

150 Equipment 3000.00

155 Vehicles 15,000.00

Total Assets: $24,555.00

Blank Blank Blank

200 Accounts Payable $50.00

250 Unearned Revenue 270.00

280 Notes Payable 10,000.00

Blank Total Liabilities: $10,320.00

Blank Blank Blank

300 MyYardBusiness $14,235.00

Total Equity: $14,235.00

What percentage of your business is Cash compared to the Total Assets of your business? Round your answer to two decimal places.

Answer: Cash/Total Assets = $6355.00/$24,555.00 = 0.25880676 = 25.88%. Cash

is more than 25% of your Total Assets.


67

Calculating Multiple Increases and/or Decreases to a Quantity when the Percents are Known Sometimes we may wish to increase a quantity by some percent, and then increase it by some other percent, or decrease it by some percent, and then increase it by some other percent. We will use a short-cut method for solving such problems using a single equation as follows: Short Cut for Calculating Multiple Increases or Decreases to a Quantity

If a quantity increases by 30%, multiply it by 1.3, since 100% + 30% = 1+0.3. = 1.3

If a quantity increases by 200%, multiply it by 3.00, since 100% + 200% = 1+2 = 3.

If a quantity decreases by 40%, multiply it by 0.60, since 100% - 40% = 1 – 0.4 = 0.6.

If a quantity decreases by 2%, multiply it by 0.98, since 100% - 2% = 1 – 0.02 = 0.98.

If a quantity increases by 100%, multiply it by 2, since 100% + 100% = 1 + 1 = 2. The quantity doubles.

If a quantity increases by 200%, multiply it by 3, since 100% + 200% = 1 + 2 = 3. The quantity triples.

To increase a quantity by a percent, multiply it by 1 plus the percent of increase expressed as a decimal. To decrease a quantity by a percent, multiply it by 1 minus the percent of decrease expressed as a decimal. The following examples illustrate that percent changes occurring one after the other are not the same as percent changes that occur simultaneously. Example 19. Compare the result of getting a 20% raise followed by a 10% raise with

the result of getting a single 30% raise. Assume the original salary is $30,000.

Answer: (30,000)(1 + 0.20)(1 + 0.10) = $39,600 (one after the other)

(30,000)(1 + 0.30) = $39,000 (simultaneous)

The difference between getting both raises simultaneously and getting them one after the other is $600.


68

Example 20. Reduce 600 by 40%, increase the result by 20%, and then decrease the result of that by 10%.

Answer: (600) (0.6) (1.2) (0.9) = 388.8

Example 21. Suppose you earn $20,000, and then receive a 2% raise, followed by a 1% raise, followed by a 5% raise, followed by a 3% raise. How much will you earn after the last raise?

Answer: (old salary)(1.02)(1.01)(1.05)(1.03) = new salary

(20,000) (1.02) (1.01) (1.05) (1.03) = $22,283.23

Example 22. If you reduce 10,000 by 5%, then reduce the result by 4%, then reduce the result by 10%, then reduce the result by 6%, what is the final answer?

Answer: (10,000) (0.95) (0.96) (0.90) (0.94) = 7715.52

Example 23. Which is better, to receive a 10% raise and then a 10% salary cut, or to receive a 10% salary cut followed by a 10% raise?

Answer: It makes no difference.

Suppose your salary is $1000.

Raise followed by cut: (1000) (1.10) (0.90) = $990

Cut followed by raise: (1000) (0.90) (1.10) = $990

Note also that the raise and the cut do not cancel each other out even though they are the same percent. You always end up receiving less than your original salary if the raise and the cut are the same percent. In order for the raise and the cut to cancel out, the percent for the raise must be greater than the percent for the cut.


69

To illustrate this, consider Examples 24 and 25.

Example 24. 11% raise followed by a 10% cut

Answer: Assuming a starting salary of $1000,

(1,000) (1.11) (0.90) = $999 You almost break even.

Example 25. 10% cut followed by 11% raise

Answer: Assuming a starting salary of $1000,

(1,000) (0.90) (1.11) = $999 Same result.

Example 26. The temperature is now 60o. If the temperature increases by 20% and then decreases by 20%, what will be the temperature?

Answer: The temperature after the 20% increase will be (60)(1.2) = 72o.

The temperature after the 20% decrease will be (72)(0.8) = 57.6o.

Once again, we see that although the percent of decrease and the percent of increase are equal, the answer is not the number with which we started. We started with 600 and ended up with 57.60

To restate: When percent changes are applied to a quantity one after the other, the result is not the same as when the percent changes are all applied at the same time. However, the order in which percent changes are done makes no difference. An increase of a number by a certain percent followed by a decrease of the result by the same percent will not yield the original number.


Section 1.5 Problems

70

1. Change 35% to a decimal.

2. Change 1.5% to a decimal.

3. Change 0.6% to a decimal.

4. Change 40% to a common fraction.

5. Change 1.8 to a percent.



8. Tripling a number is the same as multiplying by what percent?

9. Write 1

%6

as a common

fraction. 10. Change 0.07% into a decimal.

11. Dividing a number by 5 is the same as multiplying it by what percent?

12. Change 1

400 to a percent.

13. Find 20% of 180.

14. Find 0.25% of 600.

15. Find 0.085% of 52,000.

16. Find 2

%3

of 60,000.

17. Find 500% of 90.

18. Human bones make up 18% of a person’s total body weight. How much do the bones of a 130-pound person weigh?

19. Muscles make up about 40% of

a person’s total body weight. How much do the muscles of a 150-pound person weigh?

20. Joe’s weight is 85% of Bill’s

weight. If Joe weighs 204 pounds, how much does Bill weigh?

21. Suppose that 2% of the people

living in a city own Cadillacs. If 6000 people own Cadillacs, how many people live in the city?

22. 390 is what percent of 600?

23. What percent of 180 is 450?

24. If you got 48 questions correct on a test, and this was a grade of 60%, how many questions were on the test?

25. If you wish to make a grade of

80% on a test consisting of 150 questions, how many must you get correct?

26. If you get 168 questions

correct on a test consisting of 200 questions, what will be your percent grade on the test?



71

27. Referring to the table of

Example 18, since Liabilities are what you owe to others and Equity is what you own of your Assets, what percentage of the business do you own?

28. If the net profit of your

business is $16,667, and Revenue is $28,566, what percentage is the Net Profit of your Revenue?

29. If your Gas Expense is $932,

and your Total Expense is $11,899, what percentage is the Gas Expense of your total Expense?

30. The following percents came

from the 24 Jan.1994, issue of U.S. News and World Report. Every year approximately 8,000 Americans suffer spinal cord injuries from the following causes: Vehicular accidents 45% Falls 22% Acts of violence 16% Sports 13% Other 4%

Find the approximate number of people in each of these categories.

31. Table salt is 40% sodium by

weight. If a box of table salt weighs 30 oz, how many ounces of sodium are in the box?

32. The U.S. Department of

Health and Human Services lists all of the different blood types and the percentages of the U.S. population with these blood types below:

Blood type Percent

A + 34%

A - 6%

B+ 10%

B - 2%

AB+ 4%

AB- 1%

O+

O - 6%

Find the percent of the population having type O+ blood.

33. The human body normally

contains 208 bones. The finger and toe bones total 56 phalanges. What percent of the bones in the body are phalanges?

34. A patient’s bill at a hospital is

$5813. If the patient’s insurance pays 80% of the bill, how much does the insurance pay?

35. It costs a lab $52.86 to

administer a blood test. If the lab adds 12% to this cost, what is the total that the lab charges?



72

36. Change 5

8 into a percent.

37. Change 0.04% into a decimal number.

38. Find 6% of 5000.

39. What is 28% of 6000?

40. Take 0.5% of 840.

41. Calculate the number that is 22% of 550.

42. How much is 70% of 450?

43. Thirty percent of 900 is how much?

44. Eighty percent of what

number is 640? 45. Six is 0.3% of what number?

46. Twenty five is what percent of 800?

47. What percent of 400 is 60?

48. What number is 1

%4

of 1200?

49. Convert 1

200 into a percent.

50. Change 0.055% into a decimal number.

51. Change 0.055 into a percent.

52. Change 13

16 into a percent.

53. Change 4

%5

into a decimal

number.

54. Change 4

5 into a decimal

number. 55. Joe earns $24 per hour. Sam

earns $20 per hour. Joe's hourly wage is what percent of Sam's hourly wage?

56. Tawan has an annual salary of

$45,000. His salary is 150% of Sheree's salary. What is Sheree's salary?

57. Pablo's weight is 75% of Raoul's

weight. Raoul weighs 180 pounds. How much does Pablo weigh?

58. Write 100 as a percent.

59. Write 100% as a number without the percent symbol.

60. An item costing $25 is

discounted by 10% and then further discounted by 40%. What is the final cost of the item?

61. Reduce 1000 by 20%, increase

the result by 15%, and then decrease the result by 10%. What is the final amount?



73

62. Which is better, to receive a 5% raise and then a 5% salary cut, or to receive a 5% salary cut followed by a 5% raise?

63. Pam spent 10% of her clinical hours in Labor and Delivery, and 15% of her

clinical hours in the ER. Her clinical hours totaled 1020.

a. How many hours were spent in Labor and Delivery?

b. How many hours were spent in the ER?

64. John missed 8% of his work days out of a 30-day period. How many days did he miss?

65. A patient who weighs 200 lbs has a 5 lb tumor removed during surgery. What

percent of the person’s total preoperative body weight was tumor?


74

1. Move the decimal two places to the left to get 0.35.

2. 0.015

3. 0.006

4. Replace % by 1

100 and

multiply to give 40 2

100 5.

5. Move the decimal two places

to the right and add the percent sign to get 180%.

6. 17%

7. 0.03%

8. Tripling means multiplying by 3 and 3 is 300%.

9. 1

600

10. 0.0007

11. Dividing by 5 is the same as

multiplying by 1

5 which is the

same as 0.2 or 20%.

12. 1

0.0025 0.25%400

13. (0.20)(180) = 36

14. 0.25% 0.0025 (0.0025)(600) = 1.5

15. 44.2

16. Replace % by 1

100 and

multiply the result times 60,000. You should get

260,000

300 which is equal

to 400. 17. 500% x 90 = 5 x 90 = 450.

18. 23.4 pounds

19. 60 pounds

20. Solve 0.85x = 204 to get x = 240 pounds

21. Solve 0.02x = 6000 to get x = 300,000

22. 65%

23. 250%

24. 60% times the number of questions is equal to 48. 0.6x = 48

48

x 800.6

25. (0.8)(150) = x 120 = x

26. x 200 168

168

x 0.84 84%200

27. Equity/Assets = $14,235/$24,555 = 0.5797189 = 57.97189%.


75

28. $16,667/$28,566 = .5834558 = 58.346%.

29. $932/$11,899 = .0783259 =

7.833%. 30. Multiply each percent by

8000. 31. 12 oz

32. Add all of the percents given and then subtract from 100% to get 37%.

33. 27%

34. $4650.40

35. $59.20

36. 5

0.625 62.5%8

37. 0.04% 0.0004

38. x = (6%)(5000) = (0.06)(5000) = 300

39. 1680

40. (0.5%)(840) = (0.005)(840) = 4.2

41. 121

42. (70%)(450) = (0.70)(450) = 315

43. 270

44. (0.80)(x) = 640

640

x 8000.80

45. 6 = (0.003)(x)

6

x0.003

2000 = x

46. 25 = (x)(800)

25

x800

0.03125 = x 3.125% = x

47. (x)(400) = 60

60

x 0.15400

x = 15%

48.

1x % 1200

4

= (0.25%)(1200) = (0.002 5)(1200) = 3

49. 1

0.005 0.5%200

50. 0.00055 51. 5.5% 52. 81.25% 53. 0.008 54. 0.8 55. 120% 56. $30,000 57. 135 pounds 58. 10,000% 59. 1 60. $13.50 61. 828 62. no difference 63. a. 102 b. 153 64. 2.4 days 65. 2.5%

Unit 1 Calculator Usage and Equation Solving Section 1.6 Solving literal equations

77

Objectives for this section Solve a linear equation containing more than one letter for a stated variable

A literal equation, sometimes called a formula, is an equation which contains letters other than the one for which you are solving. The steps involved in solving such equations are exactly the same as those involved in solving equations containing only a single letter. The solution to a literal equation may contain both numbers and letters. You must specify which letter is the unknown in order to solve a literal equation.

Example 1. Solve 20x + 5y = 100 for the letter y.

Answer: 5y = 100 - 20x Subtract 20x from both sides

y = 20 - 4x Divide both sides by 5

Example 2. Solve a

kx

for the letter x.

Answer: a = kx Multiply both sides by x

a

xk

Divide both sides by k

Notice that the letter you are finding may appear on either side of the equal sign. In the final answer, the letter you are finding should appear by itself and only once, and it must not appear in the bottom (denominator) of a fraction. Also, the variable you are finding must not have a minus sign on it in your final answer.

Example 3. Solve the equation 8x - y = 2x + 4d for the letter x.

Answer: 6x - y = 4d Subtract 2x from both sides

6x = 4d + y Add y to both sides

4d yx=

6 Divide both sides by 6


78

Example 4. Solve r t + m = d for the letter t.

Answer: r t = d – m Subtract m from both sides

d mt=

r Divide both sides by r

Example 5. Solve 5y = 2y + k for the letter y.

Answer: 5y - 2y = k Subtract 2y from both sides

3y = k Combine like terms

k

y3

Divide both sides by 3

Example 6. Solve p + e = k – w for the letter p.

Answer: p = k - w - e Subtract e from both sides

Example 7. Solve pv = nrt for the letter t.

Answer: pv

tnr

Divide both sides by nr

Example 8. Solve p v n r t for the letter v. (This is a formula from chemistry.)

Answer: nrt

vp

Divide both sides by p

Example 9. Solve I = PRT for the letter R. (This is the simple interest formula.)

Answer: I

RPT

Divide both sides by PT


79

Example 10. Solve P = 2L + 2W for the letter L.

(This is the formula for the perimeter of a rectangle.)

Answer: P - 2W = 2L Subtract 2W from both sides

-P 2W

L2

Divide both sides by 2

Subscripts and Capitalization

Literal equations may contain both upper case (capital) letters and lower case (small) letters. You should treat the upper case of a letter and the lower case of the same letter as completely different letters. Observe the following example: Example 11. Solve A - a = B + b for the letter A.

Answer: A = B + b + a Adding a to both sides

Note that B and b are treated as if they are completely different letters. They are not like terms, and so are not added together in the final answer. A subscript is a small number appearing at the bottom right hand side of a variable,

such as 3b (read this as “b sub 3”) or 5a (read this as "a sub 5"). Variables with

different subscripts are treated as if they were different letters.

Example 12. Solve 2 1 2 1y y m(x x ) for 2x . (This is the equation of a line.)

Answer: 2 1 2 1y y mx mx Multiplying out parentheses

2 1 1 2y y mx mx Adding 1mx to both sides

-2 1 12

y y mxx

m Dividing both sides by m

Example 13. Solve 1 1 2A a a A a for the letter A.

Answer: 2 1 1A a a a A Subtracting a, 1a , and 1A from

both sides. Notice how none of these letters may be combined because of capitalization and subscripts.


80

Solve for the letter indicated:

1. 10x + 5y = 60 for y

2. 12x – 4y = 16 for y

3. 10y – 80x = 400 for y

4. 5x + 2y = 7 for y

5. I = PRT for P

6. I = PRT for R

7. I = PRT for T

8. A = P + I for P

9. A = P + I for I

10. C 2 r for r

11. r = k – d for d

12. mc = xb for c

13. 5x – 2y = 8 for y

14. sp + e = 15x + v for x

15. 5x – 8y = 15x + v for x

16. 1

A bh2

for b

17. x

d5

for x

18. y – e = f for y

19. vw + tx = ab for x

20. a (x + r) = t for x (Hint: multiply out parentheses first)

21. x

ea

for x

22. ax = efgh for x

23.

(n T)R

A P 1n

for P

24. 2 1x B x b x for 1x

25.

1 2b bA h

2 for 1b


81

26. The concentration of an enzyme is given by:

t

s

AVC

fV

where: A = absorbance as read on a spectrometer

tV = total volume

sV = sample volume

f = a constant for the substance

Solve this equation for total volume, t

V .

27. Auditors are concerned with “taxable measure” of a business when ownership is transferred. The formula they use is:

P(C L)M

T

where: M = taxable measure

P= taxable personal property

C= cash

L = liabilities assumed

T = total consideration

a) Solve this equation for T , total consideration.

b) Solve this equation for P , personal property.


82

1. 5y = 60 – 10x y = 12 – 2x

2. -4y = 16 – 12x y = -4 + 3x

3. y = 8x + 40

4. y = -2.5x + 3.5

5. I

PRT

6.

IR

PT

7.

IT

PR

8. A – I = P

9. A – P = I

10.

Cr

2

11. r + d = k

d = k – r

12. xb

cm

13. 5x = 8 + 2y 5x – 8 = 2y

5x 8y

2

14. sp + e – v = 15x

sp e vx

15

15. 5x – 8y = 15x + v -8y = 15x + v - 5x -8y = 10x + v -8y – v = 10x

8y vx

10

16. 2A = bh

2A

bh

17. x = 5d

18. y = f + e

19. tx = ab – vw

ab vwx

t

20. ax + ar = t ax = t – ar

t arx

a

21. x = ae

22.

efghx

a all that is needed is division


83

23.

(n T)

AP

R1

n

It looks complicated, but all you have to do is to divide.

24.

21

x B xx

b

25.

1 2

1 2

2 1

21

2A (b b ) h

2A b h b h

2A b h b h

2A b hb

h

26. s

t

fV CV

A

27. a)

P(C L)T

M

b)

MTP

C L

Unit 1 Calculator Usage and Equation Solving Section 1.7 Unit review problems

85

No formulas or notes will be given or allowed on the Unit 1 Test.

1. Calculate the following, and give your answer in scientific notation rounded to the nearest tenth.

a) 24 28 192.3 10 67 2.8 10

b)

4

5 6

2.58 1363.8

4.88 10 1.37 10

c) 4.67 – (-2.35)(2.17 – (-1.84))

2. Round off correctly.

a) 58.464 999 999 to the nearest hundredth

b) 4,542,688,716 to the nearest hundred million

c) 784,265,668,913.456 987 213 to the nearest thousandth

d) 13 hours 34 minutes to the nearest hour

e) 13 pounds 6 ounces to the nearest pound

f) 657 to the nearest 5.

g) 953 to the nearest 5.

h) For the following diagram, round the measurement to the nearest 1

4 inch.


86

3. Calculate and round off correctly according to the rules for rounding measured numbers.

a) 13.458 + 9.425 6 – 0.036 79

b) 0.000 058 723 - 45.7

4. a) Change 134.66 10 into standard notation.

b) Change 0.000 000 000 000 000 000 000 087 39 into scientific notation.

5. Multiply and simplify.

a) 5x + 2x + x + 3

b) 4(3x – y) – 2(x – 3y)

6. Solve each of the following for the variable indicated.

a) 2.3x + 4.7 = 42.8 for x

b) 10(x – 5) = -3(x + 4) for x

c) a(x – d) = e for x

d) x

350

for x

e) 50

4x

for x

f) 20x + 5y = 600 for y

g)

x 40

1.25

for x

h) x + 0.3x = 104 for x

i) x – 0.25x = 180 for x

j)

200 x

0.550

for x

k) 200 + 0.6x = x for x

l) I = PRT for T

m) x 5

24 8 for x

n) 3 60

7 x for x


87

7. Set up the equation, and then solve it.

a) The sum of seven times a number and 20 is 440. Find the number.

b) The sum of two consecutive integers is 361. Find the two integers.

c) The sum of two consecutive even integers is 466. Find the two integers.

d) If the Assets of a company are $135,000 and the Equity is $96,000, what are the Liabilities?

8. Convert 3

5 into a percent.

9. Convert 2.7 into a percent.

10. Convert 1

%8

into a decimal.

11. Convert 500% into a decimal.

12. Fourteen hundred is what percent of four hundred?

13. If you get 225 questions correct on a test consisting of 300 questions, what will be your percent grade on the test?

14. If your Advertising Expense is $535 and your Total Expenses is $7220, what

percentage is the Advertising Expense to the Total Expenses? Round your answer to two decimal places.

15. If your small business has Revenue of $1175 and Expenses of $1200 in the

month of February, did your company earn a profit or a loss for that month? 16. If your small business has Revenue of $5600 and Expenses of $1575 in the

month of October, did your company earn a profit or a loss for that month?


88

17. The following table shows the Revenue and Expenses for your small business for the 4th quarter of the year. Did your business earn a profit or a loss for that quarter?

Oct $5600 $1575

Nov $1200 $1600

Dec $1000 $1200

Total $7800 $4375

18. If the Equity in a small business at the end of 2013 was $23,500, and the Profit

for 2014 was $13,700, what is the total Equity at the end of 2014? 19. Additional Problems

a) Calculate 2 2 26(3.6 8.4) 5(6.3 8.4) 9(13.9 8.4)

20

b) Find 75% of 2000.

c) What percent of 200 is 25?

d) Calculate 2.8 + 0.123 + 21.72 + 113.0 and round correctly according to the rules for rounding measured numbers.

e) Change 5.692 3 x 10-9 into ordinary notation.

f) Multiply and simplify 4(2x + 5y + 6) + 3(x – 2y + 1).

g) Solve 5x + 2y = 20 for y.

h) Solve x + 3x + 2x + 20 = 180.

i) Solve x – 0.35x = 1300.

j) Solve

x 180

1.340

k) Find a number if five times the number plus 30 is equal to 70.


89

1. a) 1.9 x 1035 b) 4.4 x 10-5 c) 1.4 x 101

2. a) 58.46 b) 4,500,000,000 or 4.5 x 109 c) 784,265,668,913.457 d) 14 hours e) 13 pounds f) 655 g) 955

h) The line segment is

112

16

inches. Rounded to the

nearest 1

4 inch, it is

122

16 or

32

4 inches.equals 2.75 inches

as a decimal.

3. a) 22.847

b) -45.7

4. a) 46,600,000,000,000

b) 238.739 10

5. a) 8x + 3 b) 10x + 2y

6. a) 16.6 b) 2.92

c)

e adx

a

d) x = 150 e) 12.5 f) 120 – 4x g) 46 h) 80 i) 240 j) 175 k) 500

l)

I

PR

m) 15 n) 140

7. a) 7x + 20 = 440 x = 60

b) n + n + 1 = 361 n = 180 n+1 = 181

c) n + n + 2 = 466 n = 232 n + 2 = 234

d) Assets = Liabilities + Equity $135000 = L + $96,000 $39,000 = L

8. 60% 9. 270% 10. 0.00125 11. 5 12. 350% 13. 75% 14. 7.41% 15. $1175 - $1200 = -$25. Loss 16. $5600 - $1575 = $4025 Profit 17. $7800 - $4375 = $3425 Profit 18. $37,200

19. a) 4.65 b) 1500 c) 12.5% d) 137.6 e) 0.000 000 005 692 3 f) 11x + 14y + 27 g) y = 10 – 2.5x

h)

80x

3

i) 2000 j) 232 k) 8

91

UNIT 2

TOPICS IN ALGEBRA AND STATISTICS

Unit 2 Topics in Algebra and Statistics Section 2.1 Coordinates and graphs

93

Objectives for this section Find coordinates of given points Draw points with given coordinates Identify quadrant containing given point Draw the graph of a line from its equation Find the intercepts of a line Interpret graphs Determine whether a point satisfies an equation

The Cartesian (Rectangular) Coordinate System

We all know that locations such as mile markers on the highway or places on maps or street addresses may be specified by numbers. We locate points on a number line by assigning a number to each point on the line. One point is arbitrarily labeled with a 0 and another point with a 1. All other points are labeled accordingly as shown on the line below, using equal spacing between every pair of consecutive whole numbers:

Generally, on a horizontal number line, the positive numbers are to the right of zero, and the negative numbers are to the left of zero. On a vertical number line, the positive numbers are above zero, and the negative numbers are below zero as shown to the right: Positions in two dimensions, such as points on a computer screen or locations on a map require two intersecting number lines, one to show the horizontal location (left and right) and the other to show the vertical location (up and down) as shown below:


94

Often on a map one of the number lines will be labeled with letters rather than numbers, and the other number line will be labeled with positive numbers. Some examples of point location on such a map are shown in the next four examples: Example 1. Write the letter A at location E3 on the coordinate system which follows:

Answer: See the coordinate system below.

Example 2. Write the letter B at location C9 on the following coordinate system: Answer: See the coordinate system below.

Example 3: What is the location of the triangle?

Answer: L6

Example 4. What is the location of the circle?

Answer: C11

The two intersecting number lines are called the coordinate axes, and the two numbers used to locate a point are called the coordinates of the point. The first coordinate gives the horizontal location of the point, while the second coordinate gives the vertical location of the point. The two axes cross at the point (0,0), which is called the origin. Such a system for labeling is called a Cartesian coordinate system. Some examples of this system follow:


95

Example 5. Place the letter A on the point (-4, 2).

Answer: See picture below.

Example 6. Place the letter B on the point (0, 3).


Example 7. Place the letter C on the point (-1, -2).


Example 8. Place the letter D on the point (3, 4).


Example 9. Place the letter E on the point (2, -1).


Example 10. Place the letter F on the point (-2, 0).



96

Quadrants are sometimes used to show the general location of points. As shown in the figure below, the upper right hand region is referred to as Quadrant I or the first quadrant; the upper left hand region is referred to as Quadrant II or the second quadrant; and so forth. Example 11. Using the figure below, determine in what quadrant the point (-5, 3) is

located? Answer: II (2nd quadrant)

Example 12. In what quadrant is the point (5, -4) located?

Answer: IV (fourth quadrant)

Example 13. In what quadrant is the point (0, 3) located?

Answer: None. Points on the axes are not in any quadrant. They are between quadrants.

Example 14. In what quadrant is the point (-7, 0) located?

Answer: None. See the previous example.


97

Each of the coordinate axes may be named by a letter. The most commonly used letter for the horizontal axis is the letter X, and the most commonly used letter for the vertical axis is the letter Y. The axes are then drawn as shown on the following coordinate system:

We now call the horizontal axis the x-axis, and the vertical axis the y-axis. However, other letters are used in various applications. Once the axes are labeled as shown, the first coordinate of a point is called the x-coordinate and the second coordinate is called the y-coordinate. Using this system for the point (7, 2), the x-coordinate is 7, and the y-coordinate is 2. Graphs of Equations

Using a coordinate system with the horizontal axis labeled with the letter x and the vertical axis labeled with the letter y allows graphing of equations involving the letters x and y. We sometimes call the x-coordinate the input for the equation, and the y-coordinate the output of the equation. To graph an equation means to graph all points (x,y) for which the equation is true. Example 15. For the equation y = 5 - 2x, find y when x is 2.

Answer: Replacing x with the number 2 gives the result y = 5 - 2(2) or y = 1. In other words, the point (2, 1) is on the graph of the equation y = 5 - 2x.


98

Example 16. For the equation 5x - 2y = 40, find y when x = 10.

Answer: Replacing x with 10 gives 5(10) - 2y = 40. Solving for y gives 50 - 40 = 2y or y = 5. In other words, the point (10, 5) is on the graph of the equation 5x - 2y = 40.

By replacing the x value and the y value of an ordered pair in the equation, we can determine if the point satisfies the equation. To satisfy an equation is to find a

solution of the equation. If the x and y values of the point are a solution to the equation, then the point is on the graph of the equation. Example 17. For 3x + 2y = 12, determine if the point (2, 3) satisfies the equation.

Answer: Replacing x with 2 and y with 3 gives (3)(2) + (2)(3) = 12. Simplifying gives 6 + 6 =12. Since 12 = 12 is true, the point (2, 3) satisfies the equation 3x + 2y = 12, and it is on the graph of the equation.

Example 18. For 3x + 2y = 12, determine if the point (1, -3) satisfies the equation. Answer: Replacing x with 1 and y with -3 gives (3)(1) + (2)(-3) = 12. Simplifying

gives 3 – 6 = 12. Since -3 ≠ 12, the point (1, -3) does not satisfy the equation. Therefore, the point (1, -3) is not on the graph of the equation.

Example 19. For x y

52 4

, determine if the point (5, 10) satisfies the equation.

Answer: Replacing x with 5 and y with 10 gives 5 10

52 4

. Simplifying gives

10 10

54 4

, 20

5, 5 54

. Since 5 = 5 is true, the point (5, 10)

satisfies the equation.


99

The way to graph an equation is a) select some x values b) use these x values and the equation to find the y values c) plot the points you have found d) draw a line through the plotted points

It may be more convenient in some cases to select some y values first. For this course, all of the equations we graph will be straight lines. However, not all graphs

are straight lines. For instance, the graph of 2y x is U shaped and is called a

parabola. Intercepts

Two convenient points to use for graphing a line are the intercepts of the line. The y-intercept is found by replacing x with 0 and finding y. The x-intercept is found by replacing y with 0 and finding x. The y-intercept is the point where the line crosses the y-axis, and the x-intercept is the point where the line crosses the x-axis. Example 20. Find the intercepts of the line 3x + 8y = 48.

Answer: Replacing x with 0 yields 3(0) + 8y = 48. So y = 6 and the y-intercept is the point (0, 6). Replacing y with 0 yields 3x + 8(0) = 48. So x = 16 and the x-intercept is the point (16, 0).


100

Example 21. Graph the equation y = 3x - 4 on the axes below.

Answer: Replacing x with 0 in the equation gives y = -4. Replacing x with 2 gives y = 2. Replacing x with 3 gives y = 5. Thus, we must graph the corresponding points (0, -4), (2, 2), and (3, 5) and then draw a line through them. The x values must be selected intelligently so that the resulting points are not located out of the region covered by your coordinate axes. The three points and the line through them are shown below:


101

Example 22. Graph the equation y = x.

Answer: This means to graph every point whose y-coordinate has the same value as its x-coordinate. Such a graph would consist of points such as (-3, -3), (0, 0), (1, 1), (4, 4), and all other points where the y-coordinate is equal to the x-coordinate. The graph is shown below:

Example 23. Graph the equation y = 2x – 1.

Answer: This means to graph those points whose y-coordinate is 1 less than twice the x-coordinate. A few points on this graph would be (0, -1), (3, 5), (100, 199), and so on. The graph is shown below:


102

Example 24. Graph the equation 2

y x 13

.

Answer: This means to graph those points whose y-coordinate is 1 more than two-thirds the x-coordinate. A good idea would be to plot the point where x = 0 and y = 1, and then points whose x values are multiples of three. For example, when x = 3, y = 3; when x = -3, y = -1; when x = 6, y = 5. Plot the points (0, 1), (3, 3), (-3, -1), and (6, 5). The graph is shown below:


103

Example 25. Graph the equation

1

y x 14

.

Answer: When x = 0, y = 1. Therefore, the y-intercept is the point (0, 1). When y = 0, x = 4. Therefore, the x-intercept is (4, 0). The graph is shown below:

Example 26. Graph the equation x 4 . This equation means that y can be any value, but x is always 4. Therefore, a few points on this line are (4, 0), (4, -2), (4, 2), and (4, 5).


104

Example 27. Graph the equation y 4 . This equation means that x can be any

value, but y is always 4. Therefore, a few points on this line are (0, 4), (-2, 4), (2, 4), and (5, 4).

Applications of Graphs

The coordinates of a point may represent something other than a location in two dimensions. For instance, the first coordinate might represent the year, and the second coordinate might represent the number of units sold by a business during that year. Thus, the point (1992, 300,000) would tell you that the business sold 300,000 units during the year 1992. Or, in another case, the first coordinate might be your average speed on a trip, and the second coordinate might be your gasoline mileage. Here, the point (68, 29) would convey the information that when you averaged 68 mph on a trip, your gasoline mileage was 29 mpg. Sometimes in cases like this, one or both of the axes are drawn with only positive numbers on them, since none of the quantities considered could be negative. In other words, the graph will often lie entirely in the first quadrant.


105

Example 28. Suppose your gasoline mileage is represented by y, and your average

speed is represented by x. If y = 35 - 0.1x gives the relationship between your gasoline mileage and your average speed for speeds of 30 miles per hour or more, find your gasoline mileage when you average 60 miles per hour.

Answer: Replacing x with 60 yields y = 35 - 0.1(60) or y = 29 mpg. Your

gasoline mileage will be 29 miles per gallon when you average 60 miles per hour.

Example 29. Graph the relationship in the previous example.

Answer: Since we know that the equation is only true for speeds of 30 mph and above, we find the value of y when x is 30. We get y = 35 - 0.1(30) = 35 - 3 = 32. This corresponds to the point (30, 32). From the previous example we have the point (60, 29). Finally, graph the two points as shown below and connect them with a line. The points are shown on the graph:

Unit 2 Topics in Algebra and Statistics Section 2.1 Problems

106

1. Plot each of the following points and write the quadrant in which they are located: A = (0, 4) B = (-0.5, 3) C = (4, 6) D = (-2, 0.6) E = (-7, 0)

F = (4, 4) G = (1, -3) H = (0, 5) I = (0.5, 0.5)

Quadrants: A___ B___ C___ D___ E___ F___ G___ H___ I___

2. Suppose I start at the origin, then move 8 units to the right, then move 5 units up, then move 10 units to the left, and finally move 2 units down. What will be the coordinates of my final destination?


107

3. Give the coordinates of each of the points below:

A______ B______ C_______ D______ E______ F______ G______

4. Circle the ordered pairs that satisfy the given equation. The answer may be none, one, two, three, or all four points.

a) 4x + y = 0 (1, -3) (2, 2) (-1, 4) (0, 0)

b) x

2y 53

(0, 2.5) (15, 0) (3, 2) (6, 6)

c) 7y = 2x - 6 (1, 1) (10, 2) (3, 0) (7, 2)

d) x + 2y = 10 (2, 4) (3, 3) (0, 5) (10, 0)

e) x - 3y = 8 (4, 4) (2, 3) (-1, 4) (11, 1)

f) y = 3 (3, 0) (0, 3) (3, 5) (5, 3)

g) x y

12 3

(-2, 6) (2, 0) (0, 3) (4, -3)

h) x = 6 (6, 0) (0, 6) (3, 6) (6, 3)

i) y = x + 7 (-7, 0) (1, 8) (9, 2) (0, -7)

j) 3x + 2y - 9 = 0 (0, 4.5) (3, 0) (0, 3) (9, 9)


108

5. a) Find the x-intercepts of the lines in problem 4.

b) Find the y-intercepts of the lines in problem 4.

6. a) Suppose that the x-coordinates of a graph could not be negative, but that the y-coordinates could be either positive or negative. In what quadrant(s) would the graph have to lie?

b) Find y when x = 2 and x and y are related by the equation y = 8 - 5x.

c) Using q as the input and p as the output, find the output of the equation p = 5q -7 when q = 2.

7. Graph each of the following by plotting three points or by finding the x and y-intercept.

a) y = x + 5


109

b) y = 3x – 8

c) y + 3x = 9


110

d) 1

y x 52

e) 3y = 2x + 4


111

f) y = 4x

g) y = -3


112

h) x = -4

i) 2x = 5y – 10


113

8. When $1000 is invested for one year in a savings account that pays simple interest, the interest (I) in dollars can be found by the equation: I = 1000r where r is the interest rate expressed as a decimal.

a) If r = 6%, find I.

b) If r = 12%, find I.

c) Draw a graph of this equation with r on the horizontal axis and I on the vertical axis.


114

9. You have been offered a new job managing a small store. Your monthly salary will be $2000 plus 20% of the company’s monthly revenue.

a) Write an equation for your monthly salary using S for salary and R for

revenue.

b) Draw a graph of this equation with revenue on the horizontal axis and salary on the vertical axis. Choose appropriate scales for the axes.

c) If the monthly revenue is $15,000, calculate your salary.

d) If your salary is $4000, calculate the monthly revenue.


115

10. The circumference (distance around) of a circle is given by C 2 r where is approximately 3.141592654. . . , and r is the radius of the circle (distance from center to circle).

a) Graph this equation with r on the horizontal axis and C on the vertical axis.

b) In which quadrant must this entire graph lie?

c) Why?


116

11. Celsius and Fahrenheit temperatures have a linear relationship. The freezing point of water is 00 C or 320 F. The boiling point of water is 1000 C or 2120 F. a) Write two ordered pairs with the Celsius temperature as the

x-coordinate using the above information.

b) Plot these two ordered pairs using 200 divisions on the Celsius axis and 400 divisions on the Fahrenheit axis.

c) Draw a line through the two points.

d) Use the line to find the Fahrenheit temperature when the Celsius temperature is 400 C.

Unit 2 Topics in Algebra and Statistics Section 2.1 Answers to selected problems

117

1. A: on y-axis

B: II C: I D: II E: on x-axis F: I G: IV H: on y-axis I: I

2. (-2,3)

3. A: (-6, 5) B: (0, 4) C: (5, 2) D: (3, -1) E: (0, -4) F: (-5, -3) G: (-3, 0)

4. a) (-1, 4) (0, 0) b) (0, 2.5) (15, 0) (3,2) c) (10, 2) (3, 0) d) (2, 4) (0, 5) (10, 0) e) (11, 1) f) (0, 3) (5, 3) g) (-2, 6) (2, 0) (0, 3) (4, -3) h) (6, 0) (6, 3) i) (-7, 0) (1, 8) j) (0, 4.5) (3, 0)

5. a) x-intercepts:

a) (0, 0) b) (15, 0) c) (3, 0) d) (10, 0) e) (8, 0) f) none g) (2, 0) h) (6, 0) i) (-7, 0) j) (3, 0)

b) y-intercepts:

a) (0, 0) b) (0, 2.5) c) (0, -6/7) d) (0, 5) e) (0, -8/3) f) (0, 3) g) (0, 3) h) none i) (0, 7) j) (0, 4.5)

6. a) I and IV b) -2 c) 3


118

7. a)

b)


119

c)

d)


120

e)

f)


121

g)

h)


122

i)

8. a) $60 b) $120

c)

Notice that the graph lies entirely in the first quadrant, since both the interest rate and the amount of interest must be positive numbers.


123

9. a) S = 2000 + 0.2R

b) Let each division on the vertical scale be $1000, and each division on the horizontal scale be $5000, for instance.

c) $5000

d) $10,000

10. a)

b) quadrant I

c) x and y are always positive in this example


124

11. a) (0, 32) (100, 212)

b) and c) see graph below

d) approximately 1040 F

Unit 2 Topics in Algebra and Statistics Section 2.2 Slope of a line

125

Objectives for this section Calculate the slope of a line using two points on the line Calculate the slope of a line using the equation of the line Calculate the slope of a line using information in an applied problem Identify whether slope is positive, negative, zero or undefined from the graph of a line Use slope to calculate the change in y when the change in x is known

Meaning of Slope

It is very useful to have a measure of the “steepness” of a line. To determine the slope of the boldfaced side of the triangle below, divide the “rise” of the line by the “run”. Rise is the vertical distance; run is the horizontal distance. This can be seen in the following example:

Slope of line = rise 5 ft

0.05run 100 ft

.

Calculating the Slope by Using Two Points on the Line

Apply the same method to a straight line. To calculate the slope of a line, start with

any two points on the line with coordinates 1 1x , y and 2 2x , y as shown below:

To calculate the rise, take the difference of the y-coordinates:

Rise = 2 1y y as shown on the graph above.

To find the run, find the difference of the x-coordinates:

Run = 2 1x x as shown on the graph above.


126

The slope of this line is given by

2 1

2 1

y yrise vertical changem

run horizontal change x x.

Remember to think of the slope as the amount that y changes when x changes by 1.

The units of the slope are units of y

units of x. That is, if y is measured in miles, and x is

measured in hours, then the slope is measured in miles

hour or miles per hour.

Find the slope of the line shown below. Pick places where the line crosses the intersection of a horizontal and a vertical grid line such as the points (-4, -1) and

(3, 4). Then compute

rise 4 ( 1) 5

run 3 ( 4) 7. Thus, the slope of the line is

5

7. This

means that y increases by 5 whenever x increases by 7, or that y increases by 5

7

whenever x increases by 1.


127

Example 1. Find the slope of the line through (3, 9) and (5, 17).

Answer:

17 9 8m

5 3 24 , which tells us that y increases by 4 units each time x

increases by 1 unit. The line looks like the picture below:

Example 2. Find the slope of the line through (-4, 2) and (1, -8).

Answer:

8 2 10m

1 4 5-2 which tells us that y decreases by 2 units each

time x increases by 1 unit. The line looks like the picture below:


Answer:

4 4 0m

8 5 30 which tells us that y doesn’t change at all when x

changes. The line is horizontal, and looks like the picture below:


Answer:

7 3 4m

2 2 0undefined . In this case, the slope of this line doesn’t

exist at all, because you cannot divide by zero. The graph of this line is vertical and looks like the picture below:


128

In summary, lines can have either positive, negative, zero, or undefined slopes.

When a line has a positive slope, as x increases, y also increases. When a line has a negative slope, as x increases, y decreases. When a line has a zero slope, as x increases, y doesn’t change at all. When a line has an undefined slope, x doesn’t change at all, and the line is vertical The horizontal change on such a line is zero, and, since we cannot divide by zero, we cannot calculate a slope for the line.

Calculating the Slope from the Equation of a Line

Suppose that a line has the equation y = 4x + 17. Thus, the points (0, 17) and (5, 37) lie on the line. Verify this!

Calculating the slope of the line yields

37 17 204

5 0 5.

Notice that the number in front of x in the equation is also 4.

The slope of a line is always the number m when the line is expressed in the form y = mx + b.

Example 5. Find the slope of the line y = -3x + 40.

Answer: Two points on the line are (2, 34) and (5, 25). Calculating the slope of

the line yields

25 34 9

5 2 3-3, which is, once again, just the number

in front of the x in the equation.


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Example 6. Find the slope of the line 15x + 3y = 60.

Answer: First solve the equation for y.

3y = -15x + 60 y = -5x + 20 (You should verify this!) So the slope is -5 with no need to find two points on the line.

Example 7. Find the slope of the line y = 70.

Answer: Write the line in the form y = 0(x) + 70. Then the slope can be seen to be zero. The line is horizontal.

Example 8. Find the slope of the line x = 1.

Answer: Since this line contains no y, its slope cannot be found. It is a vertical line. Its slope is undefined.

Example 9. Find the slope of the line 2x - 3y = 6.

Answer: The equation is in the wrong form for what is desired. Solve the equation for the letter y. -3y = -2x + 6 Subtracting 2x from both sides

2x 6y

3 Dividing both sides by -3

2

y x 23

Dividing out the right side

Note that the -3 on the bottom must be divided into both -2x and 6. The

slope is 2

3.

Some Applications and Interpretations of Slope

I. Slope may be the grade of a road.

Highway signs do not write the plus or minus sign on slope since they assume that you will know whether you are going uphill or downhill. For instance a stretch of Highway 25 north of Greenville, SC, has a sign stating: 6% grade for the next 3 miles. This means that the slope of this section of road is 0.06; or that you descend 6 feet for every 100 feet you move horizontally if you are going downhill; or that you ascend 6 feet for every 100 feet you move horizontally if you are going uphill.


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The grade of a highway is of great interest to heavy-duty truck drivers on long downhill stretches, since their brakes may overheat and fail on such grades. Example 10. If you are traveling on a road with a 5% grade and going uphill, how

many feet do you move vertically if you move forward horizontally 3000 feet?

Answer: h

0.053000

h = 0.05(3000) h = 150 feet

You move up 150 feet vertically.

II. Slope may be the pitch of a roof.

Another use of slope is the pitch of a roof. Pitch is usually given in the form 4/12 or 7/12 or 12/12 with the denominator always being 12. (This is probably because there are 12 inches in a foot.) If a house has a 5/12 roof, then the roof rises 5 feet for every 12 feet of horizontal distance. Since there are 12 inches in every foot, a 5/12 roof also rises 5 inches for every 1 foot of horizontal distance. Note that the ratio of the vertical to the horizontal is the same for both roofs.

Note: Specifications or plans for roads or roofs may specify the ratio of the run to the rise rather than the rise to the run. In this case a slope of 1/5 would be written 5/1. However, mathematically, slope is always rise divided by run.


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Example 11. If the roof below has a 5/12 pitch, find h.

Answer: h 5

18 12

12h 90

90

h12

7.5 ft

Houses built in regions with heavy snowfall often have steep roofs to allow snow to slide off. The pitch of a roof in these regions might be 12/12, which is a slope of 1. Further south a roof need not be so steep. The pitch of a roof in this area might be 5/12 or 4/12. A flat roof such as is found on commercial buildings would have a pitch of zero.

Example 12. What would be the pitch of a roof if the house was 28 feet wide, and the rise in the roof was 7 feet?

Answer: The run of the roof is half of 28 or 14 feet. The rise is 7 feet. Because

pitch always has a denominator of 12, we must find the slope that has

a denominator of 12 but at the same time is equal to the given rise

run.

Therefore, rise 7 x

run 14 12. Solving for x yields x = 6. The pitch of the

roof is 6

12.


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III. Slope may be the rate of change of y compared to x.

Example 13. Suppose it is known that 2,500,000 automobiles were sold in 1992, and 3,400,000 automobiles were sold in 1994. Find the slope of the line passing through these two points, assuming that y is the number of automobiles sold and that x is the year in which they were sold. Give the units for the slope.

Answer: rise change in automobile sales

sloperun number of yeas

3,400,000 2,500,000 900,000

1994 1992 2

automobiles450,000

year.

In other words, automobile sales are increasing at the rate of 450,000 automobiles per year. Over this two-year period, on average, 450,000 more automobiles are sold each year than in the previous year.

The number 450,000 is the rate of change of automobile sales compared to the year; that is to say, the rate of change is how much automobile sales increase each year.

Example 14. Find the rate of change of a person’s weight compared to height if the

person’s weight is 80 pounds when height is 54 inches, and weight is 128 pounds when height is 66 inches.

Answer: Change in weight is 128 - 80 = 48 pounds.

Change in height is 66 - 54 = 12 inches.

Therefore, 48 pounds

12 inches

pounds4

inch. Weight is changing at a rate of

pounds4

inch compared to height. On the average, the person’s weight

increased by 4 pounds for each inch of height gained.


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Example 15. Suppose a bucket of sand leaks 10 pounds of sand over a 5-minute

period. Find the rate of change of the amount of sand in the bucket compared to time.

Answer: The change in the amount of sand is -10, since the amount of sand

decreases by 10 pounds. The change in the time is 5 minutes. This is how long it takes the sand to leak out. The rate of change (slope) is

10

5 or -2 pounds per minute. The minus sign in front of the 2 tells

us that the amount of sand in the bucket is decreasing, rather than increasing, as time passes. If we were to draw a line with y representing the amount of sand in the bucket and x representing the time, then the slope of this line would be -2.

Example 16. Suppose that the sales of a company increase from $300,000 to

$900,000 over a 6-year period. Find the average rate of change of sales compared to time.

Answer: The rate of change is

rise $900,000 $300,000 $600,000

run 6 years 6 years

dollars100,000

year

Sales are increasing on the average at a rate of $100,000 per year.

IV. In Accounting, slope may be the variable portion of mixed costs.

Mixed Costs are costs that are partially variable and partially fixed. The variable portion depends on how much of a resource is consumed. An example of a mixed cost for an individual is an electric bill. An electric bill is composed of a Base Rate and a Variable Rate. The Base Rate is what you pay each month whether you use electricity or not. The Variable Rate is the amount you pay per kilowatt hour (KWH) of electricity you consume. To compute the variable cost rate (slope), accountants may use the high-low method. This method uses two sets of numbers: 1) the total dollars of mixed costs occurring at the highest volume of activity and 2) the total dollars of mixed costs occurring at the lowest volume of activity.


134

Example 17: ABC company has accumulated the following Shipping Expense data over a period of eight (8) quarters for the years 2013 – 2014.

ABC Company

Shipping Expense

Quarter Units Sold (X)) Expense (Y)

1st - 2013 16,000 $160,000

2nd 18,000 $175,000

3rd 23,000 $217,000

4th 19,000 $180,000

1st - 2014 17,000 $170,000

2nd 20,000 $185,000

3rd 25,000 $232,000

4th 22,000 $208,000

Totals 160,000 $1,527,000

ABC would like to estimate the variable portion of their Shipping Expenses based on the units sold as follows: 1. Find the quarter with the HIGHEST Shipping Expense (3rd quarter, 2014).

$232,000 for 25,000 Units Sold

2. Find the quarter with the LOWEST Shipping Expense (3rd quarter, 2014).

$160,000 for 16,000 Units Sold

3. Find the difference in the Shipping Expense, $232,000 - $160,000 = $72,000.

Find the difference in the Units Sold, 25,000 - 16,000 = 9000.

4. Divide the change in the Shipping Expense by the change in the Units Sold. $72,000/9000 = $8.00 per Unit sold.

You can also compute the variable cost rate using the slope formula by using the information to obtain two points in the form (Units Sold, Shipping Expense); (25,000, $232,000) and (16,000, $160,000).

Once again, a positive slope (or rate of change) means that y gets larger when x gets larger. A negative slope (or rate of change) means that y gets smaller when x gets larger. A zero slope (or rate of change) means that y does not change when x gets larger.


135

1. Find the slopes of the lines containing each of the following pairs of points:

a) (1, 6) and (0, -2) f) (3, -7) and (3, 8)

b) (0, 0) and (4, 5) g) (4, -8) and (11, 5)

c) (5, 3) and (5, -5) h) (2, 7) and (-1, 7)

d) (-6, -3) and (2, -5) i) (-6, -3) and (-2, -5)

e) (0, 1) and (1, 6) j) (4, 1) and (4, 10)

2. a) Which of the pairs in the previous problem lie on a vertical line?

b) Which of the pairs in the previous problem lie on a horizontal line?

3. Determine the slope for each of the following lines:

a) x = 5 f) x = 2y + 1

b) y = 7x + 1 g) y = 8

c) 2x - 4y = 6 h) 3y - x = 6

d) y = 11 i) 7x + 2y + 5 = 0

e) x + y = 10 j) x = 8

4. Find the y-intercepts for each of the lines in problem #3.

5. Find the x-intercepts for each of the lines in problem #3.

6. A formula for the height of a candle as a function of time is h = 12 - 2t, where h is the height of the candle in centimeters, and t is the time in hours that the candle has been burning.


136

a) Graph the equation of growth with h on the vertical axis and t on the

horizontal axis.

b) What was the initial height of the candle before it was lit?

c) How much does the height of the candle change each hour?

d) How many hours does it burn?

7. If the grade of a highway is 8%, and the highway rises 1600 feet, how far have you traveled horizontally? See picture.

8. If the pitch of your roof is 6

12, and the width of the roof is only 3 feet, what

must be the height, y, of the roof in inches? See picture.


137

9. A sign on a mobile home sales lot says “4 ft to 12 ft roof pitch.” What is the slope of the roofs on these mobile homes?

10. The number of cars in a parking lot increases from 100 to 500 over a four-hour

period. What is the rate of change of the number of cars in the parking lot with respect to the number of hours? In other words, how many additional cars are entering the parking lot per hour?

11. The number of gallons in the gas tank of a car changes from 20 gallons to 12

gallons during the same time the mileage on the odometer of the car changes from 22,000 miles to 22,240 miles. What is the rate of change of the miles traveled with respect to the amount of gasoline in the tank?

12. The number of acres planted in tobacco changed from 5000 acres to 2000 acres

over a 6-year period. What was the rate of change of the number of acres planted in tobacco with respect to the year?

13. John’s salary was $20,000 in 1990 and was still $20,000 in 1998. What is the

rate of change of his salary with respect to the number of years? 14. a) Give an example of the equation of a line that has a positive slope.

b) Give an example of the equation of a line that has a negative slope.

c) Give an example of the equation of a line that has a zero slope.

d) Give an example of the equation of a line that has an undefined slope.

15. A motor bike drives on a terrain rising 62 feet vertically over a horizontal distance of 248 feet. Find the grade of the terrain written as a percent.

16. a) Give an example of the equation of a line which has a y-intercept but

no x-intercept.

b) Give an example of the equation of a line which has an x-intercept but no y-intercept.

17. a) Give an example of the equation of a line where y decreases as x

increases. b) Give an example of the equation of a line where y increases as x increases.

c) Give an example of the equation of a line where y doesn't change as x increases.


138

18. a) Describe how you may find the slope of a line using its equation.

b) Describe how you may find the slope of a line using two points which lie on the line.

19. a) If y = 5x – 675, find the rate of change of y compared to x.

b) If 32x + 16y = 640, find the rate of change of y compared to x.

c) If y increases by 20 when x increases by 5, calculate the slope of the line.

d) If y decreases by 40 when x increases by 8, calculate the slope of the line.

20. If a piece of machinery declines in value from $45,000 when it was purchased to $20,000 five years after it was purchased, by how much does the value of the machine decline per year on the average? Find the slope of the line relating its value to its age.

21. If the value of a house which cost $130,000 rises to $160,000 three years later,

by how much does the value of the house rise per year on the average? Find the slope of the line relating the value of the house to its age.

22. If the height of a tree remains constantly 65 feet, find the slope of the line

relating the height of the tree to its age. 23. The number of people living in a small town decreases from a population of

12,000 to a population of 10,000 over a 4-year period. Find the rate of change of the population compared to time.

24. Using the following table for ABC Company, find the variable cost rate.

ABC Company Units Sold

Shipping Expense

High activity level 20,000 $85,000

Low activity level 6000 $15,000


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1. a)

2 6 88

0 1 1

b)

5 0 5

4 0 4

c)

5 3 8undefined

5 5 0

d)

5 3 5 3 2 1

2 6 2 6 8 4

e)

6 1 55

1 0 1

f)

8 7 8 7 15undefined

3 3 0 0

g)

5 8 5 8 13

11 4 7 7

h)

7 7 00

1 2 3

i)

5 3 5 3 2 1

2 ( 6) 2 6 4 2

j) 10 1 9

undefined4 4 0

2. a) c, f, j since the slopes are undefined

b) h since the slope is zero

3. a) slope = undefined (vertical line)

b) slope = 7

c) slope = 1

2

d) slope = 0

e) slope = -1

f) slope = 1

2

g) slope = 0 (horizontal line)

h) slope = 1

3

i) slope = 7

2

j) slope = undefined

4. a) none

b) (0, 1)

c)

30,

2

d) (0, 11)

e) (0, 10)


140

f)

10,

2

g) (0, 8)

h) (0, 2)

i)

50,

2

j) none

5. a) (5, 0)

b)

1, 0

7

c) (3, 0)

d) none

e) (10, 0)

f) (1, 0)

g) none

h) (-6, 0)

i)

5, 0

7

j) (8, 0)

6. a)

b) 12 cm

c) decreases by cm

2hr

d) 6 hours

7. 1600

0.08x

1600 = 0.08x

1600

x0.08

20,000 feet = x

8. 3 feet = (3)(12) = 36 in

y 6

36 12

12y = 216

216

y 1812

inches

9. 4 1

12 3


141

10.

500 100

100 cars per hour4

11.

22,240 22,000 240 miles30

12 20 8 gallon

The minus sign indicates that the amount of gasoline in the tank is decreasing.

12. 2000 5000 3000 acres

5006 6 year

The minus sign indicates that the number of acres is decreasing by 500 each year.

13.

20,000 20,000 00

1998 1990 8

His salary is not changing.

14. a) answers vary – one example y = 2x+1

b) answers vary – one example y = -2x -3

c) answers vary – one example y = 5

d) answers vary – one example x = 4

15. 62

25%248

16. a) answers vary – one example y = 5

b) answers vary – one example x = 4

17. a) answers vary – one

example y = -3x

b) answers vary – one example

y = 3x

c) answers vary – one example

y = 5

18. a) Solve the equation for y. The coefficient of x is the

slope.

b) Use the slope formula.

19. a) 5 b) -2 c) 4 d) -5

20. decreases by $5000

year

slope is -5000

21. increases $10,000

year

slope is 10,000

22. slope is 0

23. people

500year

24. $5/unit sold

Unit 2 Topics in Algebra and Statistics Section 2.3 Interpreting and Finding Equations of Lines

143

Objectives for this section

Interpret physical meaning of slope Interpret physical meaning of intercept Find the equation of line using calculator

Interpretation of Slope and y-intercept

With the exception of the vertical line, which has the equation x = k where k is a number, the equation of a line always has the form y = mx + b. Example 1. y = 6x + 200

Two important facts are contained in this equation.

a) The slope of the line is 6, which means that y increases by 6 units whenever x increases by 1 unit.

b) The y-intercept for the line is (0, 200), which means that the line crosses the y-axis at this point and that y = 200 when x = 0.

Here are some possible interpretations of the equation in the previous example.

Suppose that in the above example, y represents the cost of manufacturing x units of some product. This equation tells us that every unit manufactured raises the total cost by $6; and that even when no units are manufactured, the cost of doing business is $200. The $200 is called the fixed cost. Suppose that in the above example, y represents the height of a shrub measured in cm, and x represents the number of years since the shrub was planted. The equation tells us that the shrub is growing at the rate of 6 cm per year, and that the height of the shrub when it was planted was 200 cm. Suppose that in the above example y is the number of gallons of water in a pool, and x is the number of hours since a water supply flowing into the pool was turned on. We know that the pool is filling at the rate of 6 gallons per hour, and that the pool contained 200 gallons when the faucet was turned on. If the pool will hold 500 gallons, we can calculate how many hours it will take to fill it by solving the equation 500 = 6x + 200.


144

Example 2. y = -3x + 60

The available information is:

a) The slope is -3, which means that y decreases by 3 units each time x increases by one unit.

b) The y-intercept is the point (0, 60), which means that the line crosses the y-axis at this point.

In Example 2 if y represents the number of quarts of water in a tub, and x represents the number of minutes since a drain was opened, the equation tells us that the water is draining out at the rate of 3 quarts per minute, and that the tub contained 60 quarts of water when the drain was opened. To find how long it takes to drain the tub, we solve the equation 0 = -3x + 60. The answer is 20 minutes.

Example 3. Suppose that the monthly salary in dollars earned by a salesperson is given by the equation y = 600 + 0.15x, where x represents the total sales by that person for the month measured in dollars.

The information available here is that the salesperson earns 15% of all sales for the month, and that the salary would be $600 even with no sales for the month. Monthly sales of $6,000 would yield a salary of 600 + 0.15(6,000) or $1,500. The slope of the line is the commission rate for the salesperson.


145

Interpreting Graphs

Information may also be gained from the graph of an equation, as shown in the following examples: Example 4. Use the graph below to answer the questions that follow.

a) Find the value of y when x = 4. b) Find the value of x when y = 4. c) Find the slope of the line. d) Find the y-intercept for the line. e) Find the x-intercept for the line.

Answers: a) y = 6 b) x = 3 c) 2 d) (0, -2) e) (1, 0)


146

Example 5. Using the graph given in the previous example, assume that x represents the number of years that a company has been in business, and that y represents the total profit of the company for that year measured in hundreds of thousands of dollars. Answer the following questions. a) What portion of the graph would need to be deleted to make it

realistically describe the situation described above? b) What would be the amount of profit for the company the third year it

was in business? c) By what amount is profit increasing each year?

Answers: a) Anything in second or third quadrant b) 4 hundred thousand dollars c) 2 hundred thousand dollars

Example 6. Use the graph below to answer the questions which follow.

a) Find y when x = 5. b) Find the amount y changes when x increases by 3. c) Find the x-intercept for the line. d) Find the y-intercept for the line.

Answers: a) y = 2 b) none c) none d) (0, 2)


147

Finding the Equation of a Vertical Line or the Equation of a Horizontal Line

The lines having the simplest equations are the vertical line and the horizontal line.

Vertical Lines

The equation of any vertical line (whose slope is undefined) is x k , where k is the x-coordinate of every point on the line.

Example 7. Find the equation of the line graphed above:

Answer: Since the line is vertical and every point has x-coordinate -2, the equation is x -2 .

Example 8. Find the equation of the line through the points (2, -3) and (2, 4).

Answer: Since one of these points lies directly above the other, the line is vertical. The equation is x 2 . Note that the slope cannot be computed for this line.

Example 9. Find the equation of the line whose slope is undefined and which passes

through the point (6, 0). Answer: Because the slope is undefined, the line is vertical. The equation is

x 6 .


148

Horizontal Lines

From the section on slope, we have seen that the slope of a horizontal line is zero. Look at the graph of the horizontal line below:

If we list a few of the points that are on the horizontal line shown above such as (-4, 2), (-2, 2), (0, 2), (2, 2), (4, 2), etc., we see that every point has a y-coordinate of 2.

The equation y 2 completely describes the line, because y is 2 for any value of x.

Because this line is horizontal, the value of x can be any number, but the value of y is always 2.

The equation of any horizontal line (whose slope is zero) is y b , where b is the

y-coordinate of every point on the line.


149

Example 10. Write the equation of the line shown above:

Answer: The equation is y -3 , because every point on the line has a y-

coordinate of -3. Example 11. Write the equation of the line which has slope of zero and passes

through the point (3, 16). Answer: A slope of zero means that the line is horizontal. The line has an

equation y 16 , because it must pass through the point (3, 16).

Finding the Equation of Any Line Which is Not Vertical

Most applications of lines involve lines which are neither vertical nor horizontal. Every line which is not vertical can be written in the form

where m is the slope of the line, and the number b is the y-intercept of the line. This form of the equation of a line is called the slope-intercept equation of a line. The y-intercept is the point where the line crosses the y-axis, and has the form (0, b). The x-coordinate of the y-intercept of a graph is always zero.


150

Example 12. Find the equation of the line having slope 5 and passing through the point (0, 2).

Answer: y = 5x + b

2 = 5(0) + b replacing y with 2 and x with 0 to find b. 2 = b y = 5x + 2

Example 13. Find the equation of the line having slope 7 and passing through the point (2, 9).

Answer: y = 7x + b

9 = 7(2) + b replacing x with 2 and y with 9. 9 = 14 + b -5 = b y = 7x – 5

Example 14. Find the equation of the line with slope -4 and passing through the point (-1, 2).

Answer: y = -4x + b

2 = -4(-1) + b 2 = 4 + b -2 = b y = -4x - 2

Finding the equation of a line using the TI-30XS MultiView Calculator

Your calculator will calculate the slope and the y-intercept given any two points on the line. However, your calculator uses the letter a for the letter m. Hence, the

equation of the line is y ax b . On the TI-30XS MultiView calculator, you calculate

a and b as follows: To enter the two points (2, 7) and (5, 13) into the calculator:

1. Press (twice).

2. Arrow down to highlight 4: Clear All.


151

3. Press to clear out any data that is

left over from a previous problem.

4. Enter the x values of each point in L1 and

the y values of each point in L2 using the

arrow keys to move between the two

columns.

5. Press . (The word stat is

above the key)

6. Arrow down to highlight: 2: 2-Var Stats

7. Press

8. For xDATA highlight L1

9. Press

10. For yDATA highlight L2

11. Press

12. Arrow down to highlight CALC

13. Press

14. Arrow down several times to highlight D:

a=2

15. Arrow down once more to highlight

E: b=3

16. Once you determine the values of a and b, write the equation in the form y = ax + b. In this example, the equation of the line is y = 2x + 3.


152

Example 15. Use your calculator to find the equation of the line through the points (2, 7) and (5, 13).

Answer: y = 2x + 3

Example 16. Use your calculator to find the equation of the line that passes through the points (-2, -3) and (-4, -5).

Answer: y = x – 1

Example 17. Find the equation of the line with slope 2.5 and having a y-intercept of (0, 4).

Answer: y = 2.5x + 4

Example 18. A shrub is 20 cm tall when it is planted, and it grows 4 cm each week for the next 15 weeks. Assuming that the equation is a straight line, write the equation relating y, the height of the shrub in cm, to x, the number of weeks since it was planted.

Answer: The slope of the line is cm

4week

, since this is the amount the height

increases for each additional week of time. Since the shrub is 20 cm tall when it is planted, this means that y = 20 when x = 0. In other words, the y-intercept is (0, 20).

Using the formula y = mx + b, the equation is y = 4x + 20.

Example 19. If the value of a piece of machinery when it is purchased is $30,000, and it is worth $10,000 five years later, find the equation relating its value, y, to its age, x, assuming that the equation is a line.

Answer: When the age, x, of the machinery is 0, its value, y, is $30,000; and

when the age is 5 years, the value is $10,000. These data correspond to the points (0, 30,000) and (5, 10,000). Using the calculator, we obtain y = -4000x + 30,000.


153

Notice that the slope in example 18 is positive, since the height of the shrub increases as its age increases; whereas in example 19, the slope is negative, since the value of the machinery decreases as its age increases. Example 20. To take the previous problem further, find the value of the machinery

7 years after it is purchased. Answer: Substitute 7 into the formula for x giving

y = -4000(7) + 30,000 y = -28,000 + 30,000 y = $2000

This equation tells you that the machinery is worth $2000 seven years after it is purchased. This situation where an object loses the same amount of value each year is called straight-line depreciation.

Example 21: Referring to the following table for XYZ Company, what would be the

equation of the line?

XYZ Company Units Sold

Shipping Expense


Low activity level 5000 $35,000

Answer: Let y represent the shipping expense.

Let x represent the number of units sold. Let a represent the variable cost rate of the shipping expense. Let b represent the fixed cost of the shipping expense.

The equation of the line would be in the form y = ax + b. Thus, Total Shipping Expense = Variable Cost Rate times the number of units sold + Fixed Cost. The variable cost rate is ($75,000 - $35,000)/(15,000 – 5000) = $4/unit sold. Using the fact that when 15,000 units is sold the shipping expense is $75,000, we have $75,000 = $4.00 x 15,000 + b $75,000 = $60,000 + b $15,000= b

Equation of the line: y = 4x + 15,000.


154

1. The price of an item, y, in terms of the number of the item sold, x, is given by y = 800 - 2x. In other words, if 300 were sold, the price would be $200 each.

a) How much does the price of an item change for each additional item sold? b) What would be the price of each item if 50 items were sold?

c) For what number of items sold would the price of each item be 0?

2. Darla's yearly salary is given by the equation y = 30,000 + 1500x, where x is the number of years since 1990 and y is her salary in dollars. a) What has been her yearly raise since 1990?

b) What was her salary in 1990?

c) What was her salary in the year 2000?

3. John's weight, y, is given by the equation y = 180, where x is the number of years since 1990. a) How much weight does John gain or lose each year?

b) How much did he weigh in the year 2000, assuming the same equation was true?


155

4. The number of gallons of water in a leaking tank is given by y = 6000 - 30x, where x is the number of hours since the tank was filled.

a) How many gallons per hour are leaking?

b) How many gallons will be in the tank after 40 hours?

c) When will the tank be empty?

5. The length of a certain snake measured in centimeters is given by y = 60 + 20x, where x is the age of the snake in years.

a) How long was the snake at birth?

b) How fast is the snake growing measured in centimeters per year?

c) In how many years will the snake be 160 centimeters long?

6. The number of persons employed by a business is given by y = 900, where x is the number of years the business has existed.

a) How many employees did the business have in its first year of existence?

b) How many employees did the business have in its eighth year of existence?

c) Assuming no employee quits, retires, or dies, how many new people are

hired each year by this business?


156

7.

a) Find y when x = -5.

b) Find the slope of the line.

c) Find x when y = 6.

d) Find the x-intercept of the line.


157

8.

a) Find the y-intercept of the line.

b) Find the slope of the line.

c) Find y when x = 3.

d) Find x when y = 6.


158

9. The amaryllis grows from a bulb to a plant with flowers in just a short time. If we let y represent the height of the plant in centimeters, and x represent the number of days since the plant was purchased, then the following points lie on the graph representing the growth of the plant: (0, 4) (2, 6) (4, 8) (6, 10). a) How tall was the plant when it was purchased?

b) Graph the points below to observe what type of curve describes the growth of the plant.

c) Write the equation that describes this graph.

d) How tall will the plant be in 16 days if the present rate of growth continues?


159

10. Write the equations of the lines having the given slope and containing the given point for each of the following:

a) m = 0 (-2, 4) f) m = 5 (8, 0)

b) m = 2 (-1, 4) g) m = 1

6 (0, -2)

c) m =1

4 (2, 8) h) m = 0 (11, 14)

d) m = undefined (4, 11) i) m = 1

2 (-8, -10)

e) m = 1

3 (0, 7) j) m = undefined

1, 6

4

11. Write the equations of the lines passing through the following pairs of points: a) (-2, 5) and (3, 6) d) (6, 7) and (-2, 7)

b) (4, 8) and (4, -10) e) (0, 5) and (3, 0)

c) (1, 2) and (3, 5) f) (-2, 1) and (0, 4)

12. a) If the temperature is presently 560 F and is rising 50 F every hour, write the equation for the temperature x hours from now, using y for the temperature in 0F.

b) What will be the units of the slope of this line?

13. Mildred had a salary of $19,000 in 1989 and a salary of $23,500 in 1992. Assume that her salary (expressed in terms of the number of years since 1989) is a line.

a) Using y for her salary and x for the number of years since 1989, find the

equation for her salary.

b) What will be the units of the slope of this line?

c) According to the equation you found, what was her salary in 1997?


160

14. a) Jack notices that the graph describing the value of his automobile in terms

of the year is a straight line with a slope of dollars

1100year

. If his car is

presently worth $14,500, how much will it be worth 3 years from now?

b) If he bought the car two years ago, how much was it worth when he bought it?

c) Write an equation expressing the value of the car, y, in terms of x, the

number of years since Jack bought it. 15. Find the equation of the line passing through (2, -5) and having slope -4.

16. Find the equation of the line passing through (1, 4) and (3, 12).

17. Find the equation of the vertical line through (-7, 200).

18. Find the equation of the horizontal line through (6, 79).

19. If a tree is 12 feet tall after 2 years and 18 feet tall after 5 years, find the equation of the line describing its growth, where y is the height and x is the number of years.

20. When it is one year old, a car is worth $21,000. When it is two years old, it is

worth $18,000. Find the equation of the line describing its value, where y is the value and x is the age of the car in years.

21. A glass of water contains 9 ounces of water at 6 am. Due to evaporation, it

contains 7 ounces of water at 4 pm the same day.

a) Find the rate of evaporation measured in ounces per hour.

b) Assume the rate of evaporation remains constant. If x represents the number of hours since 6 am and y represents the amount of water in the glass, find the equation relating x and y.


161

22. Find the equation of the line, using the following table:

John Doe Company

Units Sold

Packaging Expense


Low activity level 35,000 $12,000

Writing Exercises

22. Can any pair of points on a line be used to calculate the slope of a line? Explain.

23. In the equation y = mx + b, what do the letters m and b represent?

24. Explain how to find the x-intercept for the equation y = mx + b.

25. Explain how to find the y-intercept for the line 5x + 4y =200.

26. Suppose two linear equations have the same y-intercept but one equation has an x-intercept that is one half of the x-intercept of the other equation. How do the slopes of the two lines compare?

27. Explain why the slope of a vertical line is undefined.

28. Explain how you can look at a line and tell whether its slope is positive, negative, zero, or undefined.


162

1. a) It decreases by $2.

b) $700 c) 400

2. a) $1500 b) $30,000 c) $45,000

3. a) 0 b) 180 pounds

4. a) 30 gallons per hour b) 4800 gallons c) after 200 hours

5. a) 60 cm b) 20 cm per year c) 5 years

6. a) 900 b) 900 c) 0 since the slope of the line is zero.

7. a) -6

b) 3

2

c) 3 d) (-1, 0)

8. a) (0, -2)

b) 4

3

c) -6 d) -6


163

9. a) The point (0, 4) tells you that the height is 4 cm when the time is zero.

b) Notice that points lie on a straight line. So you may use any two of them to find the equation of the line.

0

1

2

3

4

5

6

7

8

9

10

0 5 10

c) The slope is

6 41

2 0, and the line passes through the point (0, 4). Using

the formula to find the equation of the line gives y = x + 4. d) y = 16 + 4 = 20 cm

10.

a) y = 4

b) y 2x 6

c) 1 1

y x 84 2

d) x 4

e) 1

y x 73

f) y 5x 40

g) 1

y x 26

h) y 14

i) 1

y x 142

j) 1

x4

11.

a) 1 2

y x 55 5

b) x 4

c) 3 1

y x2 2

d) y 7

e) 5

y x 53

f) 3

y x 42


164

12. a) y = 5x + 56

b) degrees

houror degrees per hour

13. a) y = 1500x + 19,000

b) dollars

year or dollars per year

c) y = 1500(8) + 19,000 = 12,000 + 19,000 = $31,000

14. a) 3(-1100) + 14,500 = -3300 + 14,500 = $11,200

b) 2(1100) + 14,500 = 2200 + 14,500 = $16,700

c) y = -1100x + 16,700

15. y = -4x + 3

16. y = 4x

17. x = -7

18. y = 79

19. y = 2x + 8

20. y = -3000x + 24,000

21. a) 0.2 oz

hr

b) y = -0.2x + 9

22. y=0.2x + 5000

Unit 2 Topics in Algebra and Statistics Section 2.4 Frequency distributions and graphs

165

Objectives for this section Find frequency distribution given data Draw histogram given data or frequency distribution Draw frequency distribution for given data Interpret frequency distribution, histogram, time-series plot

Introduction to Statistics

The term “statistics” is used in more than one way.

1. A name for data, such as batting averages, number of oil spills, profits for businesses, ages of a group of people, etc., as when one sports commentator says to another, “Give me the statistics on this player.”

2. The name of the branch of mathematics which analyzes data and attempts to

summarize and draw conclusions from the data, as when a business uses a statistics program in a computer to analyze sales data and make forecasts.

3. The word “statistic” may also be used to describe a number derived from a set

of data which attempts in some manner to describe or summarize the data. One example is the mean, which we will encounter later.

Statistics are widely used in making business decisions, deciding whether a person’s test score is adequate to get a job, helping make court decisions about discrimination, determining how much government money a district gets for some purpose, etc. Generally, information is obtained from a sample of a population, rather than from the entire population. For instance, if we wish to know the opinion of Greenville Tech students on some subject, we would interview perhaps 50 or 100 students rather than the entire student body. Of course, we would have to be sure that the students we selected were truly representative of the student body by using correct surveying techniques. In statistics, the word sample means a subgroup of all of the data under consideration. The collection of all of the data under consideration would be called the population. Incidentally, the word “data” is plural not singular.


166

Frequency Distributions

One way of dealing with large amounts of data is to organize the data into groups or classes. The class may consist of just a single number or all of the data between two different numbers. The difference between the lower limit of a class and the lower limit of the class just above it is called the class width. By counting the amount of data in a class, we determine the frequency for the class. Example 1. The ages of a group of people are as follows: 18, 19, 18, 22, 18, 19, 21,

23, 22, 22, 19, 20, 21, 21, 18, 18, 19, 18, 22, 22, 22, 22, 23, 21. Since there are only six different ages, make a frequency distribution using the six ages as the classes.

Answer:

age tally frequency

18 //// / 6

19 //// 4

20 / 1

21 //// 4

22 //// // 7

23 // 2

When making a frequency distribution, every class should be the same width, and there should be between 5 and 20 classes depending upon how much data there is. Observe how a frequency distribution is constructed in the following example: Example 2. On a final exam, an instructor recorded the following grades:

83, 88, 65, 94, 43, 96, 81, 72, 70, 71, 71, 71, 99, 54, 66, 66, 87, 87, 90, 80, 70, 68, 58, 67, 67, 83, 77, 77, 77, 49, 94, 84, 84, 60, 60, 60, 50, 50, 99, 44, 40, 75

Construct a frequency distribution for the grades shown.

Grade Class Tally of Grades Frequency

40 - <50 //// 4

50 - <60 //// 4

60 - <70 //// //// 9

70 - <80 //// //// 10

80 - <90 //// //// 9

90 - <100 //// / 6


167

Possible Make each class interval ten points as follows: Answer: 40 - <50; 50 - <60; 60 - <70; 70 - <80; 80 - <90; 90 - <100

Then count the number of grades in each class and record the result in table form using tally marks to keep track of what has been counted. Add the number of tally marks for each class.

Graphical Displays of Data

A useful statistical tool for presenting the classes is the frequency histogram, which is just a type of bar graph with no spaces between the bars. The classes are shown on the horizontal axis and the bars are drawn directly above the classes. The heights of the different bars represent the amount of data in the classes below the bars. Some guidelines for histograms:

1. The classes must all be the same width, making all of the bars the same width.

2. The scale for the classes must lie on the horizontal number line (x-axis).

3. The scale for the amount of data (frequency) in each class must lie on the vertical number line (y-axis).

4. The height of a given bar represents the amount of data (frequency) in the

class shown at the bottom of that bar (classes). 5. There should be between 5 and 20 classes.

A frequency histogram may be drawn using the grade frequency distribution obtained earlier.


168

Example 3. Construct a frequency histogram from the following distribution.

Age Ranges Number of Persons in Range

21 - <26 150

26 - <31 200

31 - <36 100

36 - <41 50

41 - <46 250

46 - <51 100

Answer:

NUMBER OF PERSONS IN DIFFERENT AGE RANGES

If the value of a variable is measured at different points in time, the data are referred to as time-series data. A time-series plot (shown below) is obtained by plotting the time in which a variable is measured on the horizontal axis and the corresponding value of the variable on the vertical axis. Line segments are then drawn connecting the points.


169

Example 4. Answer the following questions using this histogram.

1. How many families does this chart represent?

Answer: Add together the heights of all of the bars 10 + 8 + 9 + 5 + 2 + 2 = 36 families

2. How many of these families have exactly 2 children?

Answer: The height of the third bar from the left which is 9 families.

3. How many of these families have more than 2 children?

Answer: Add together the heights of the three bars representing the numbers of families having either 3, 4, or 5 children. This yields 5 + 2 + 2 = 9 families.

4. What is the total number of children in all of these families?

Answer: Multiply the number of families times the number of children for each bar, and then add together the resulting numbers. The result is 10(0) + 8(1) + 9(2) + 5(3) + 2(4) +2(5) = 59 children.

0

2

4

6

8

10

12

0 1 2 3 4 5Number of Children

Number of families having given number of children


170

On a histogram the classes are numerical and are generally shown on the x-axis. The height of each bar represents the amount of data (frequency) in each class. Bar graphs which are not histograms may also be used to display data. Generally the classes are non-numerical, and the heights of the bars may or may not represent the number of data in the classes. In such cases, the bars are usually drawn with spaces between them as shown in the following examples:

Example 5. Draw a bar graph showing the number of pieces of several different colors of candy found in a package. Use the following chart.

Color Number of Pieces

Green 15

Brown 10

Red 5

Yellow 8

Blue 8

0

2

4

6

8

10

12

14

16

Green Brown Red Yellow Blue

Number of Pieces of Each Color


171

A circle graph or pie chart can also be used to convey information. Using the information contained in the frequency distribution of Example 2, a circle graph is shown below.

Grade Class Tally of Grades Frequency

40 - <50 //// 4

50 - <60 //// 4

60 - <70 //// //// 9

70 - <80 //// //// 10

80 - <90 //// //// 9

90 - <100 //// / 6

Notice that the circle graph is especially good for showing the percent or fraction of students in each class in addition to the actual number of students in each class. We would calculate the percents for the pie chart by dividing the number of students in a given grade category by the total number of students. In the case of the 90 - <100 category, this would be 6/42 = 0.143 = 14% (approximately). Since construction of circle graphs involves measurement of angles, no exercises will involve creation of such graphs. You may be asked to answers questions about information contained in circle graphs.

40 - <5010%

50 - <6010%

60 - <7021%

70 - <8024%

80 - <9021%

90 - <10014%

Percents of Students Scoring in Different Grade Ranges


172

1. The grade distribution for a Math 155 class is as follows:

Grade Frequency

A 3

B 5

C 14

D 2

F 1

Draw a frequency bar graph from the above information.

2. The following are amounts that students paid for textbooks (to the nearest dollar) during a recent summer term.

105 122 106 100 114 112 130 111 128 129 103 119 123 102 108 118 121 105 120 117 115 117 138 106 111 120 130 117 98 126

a) Construct a frequency distribution table so that the first class begins at $95 and the class interval is $5.

b) Draw a frequency histogram.

3. Twenty five states having the highest high school dropout rate in 1986 (Source: USA Today, Feb. 26, 1988) are given below. Summarize the results and represent graphically.

Washington, DC 43.2% Florida 38.0% Louisiana 37.3% Georgia 37.3% Arizona 37.0% Mississippi 36.7% New York 35.8% Texas 35.7% South Carolina 35.5% Nevada 34.8% California 33.3% Rhode Island 32.7% Alabama 32.7%

Tennessee 32.6% Michigan 32.2% Alaska 31.7% Kentucky 31.4% North Carolina 30.0% Delaware 29.3% Hawaii 29.2% Oklahoma 28.4% Indiana 28.3% New Mexico 27.7% Colorado 26.9% New Hampshire 26.7%


173

4. The circle graph below describes 242 (estimated) million automobile tires that are scrapped in the US every year. Source: US Environmental Protection Agency and national Solid Waste Management Association. Tampa Tribune, June 29, 1992.

a) Interpret the circle graph.

b) Convert the circle graph to a bar graph.

5. Below is a circle graph of Educational Attainment in 1990. Find the following based on the graph:

a) the percentage of persons with a bachelor’s degree or higher b) the percentage of persons with 1-3 years of college c) the percentage of persons with less than 4 years of high school d) the percentage of persons who are high school graduates

Dumped 77.6%

Burned for Fuel 10.7%

Recycled into New Products 6.7%

Exported 5%

Bachelor's Degree or

Higher (20.3%)

Less than 4 years High

School (24.8%)

High School Graduate (30.0%)

1-3 Years College (24.9%)


174

6. Look at the bar graph given below (Source: the South Carolina Nurse, December, 1996), and answer the following questions:

a) Where is the largest increase in employment in 1993-94? b) What is the percentage of increase in the hospital setting? c) What is the percentage of increase in the Long-term Care setting? d) Which setting had the smallest increase?

Percentage increase in RN’s by employment setting in SC, 1993-94

7. Looking at the bar graph below from The South Carolina Nurse, December, 1996, answer the following questions:

a) What degree is most common in the SC nursing workforce? b) What percentage of the SC nursing workforce has PhD’s?

Educational Mix of SC Nursing Workforce 1995

Writing Exercise:

8. For the time-series plot at the bottom of page 168, answer the question posed, “How am I doing over time?”

0

5

10

15

20

25

30

Hospital Health dept. Longterm Care Home health

0

5

10

15

20

25

30

35

LPN Dip ADN BSN MSN PhD

%


175

1.

Grade Distribut ion

0

2

4

6

8

10

12

14

16

A B C D F

Number of

Students

Making Grade

2. a) Use tally marks to keep track of the number of students in each class.

Amount Paid for Textbooks in $

Tally of Students

Frequency of Students

95 - <100 / 1

100 - <105 /// 3

105 - <110 //// 5

110 - <115 //// 4

115 - <120 //// / 6

120 - <125 ///// 5

125 - <130 /// 3

130 - <135 // 2

135 - <140 / 1

b)


176

3. Use categories of 25 - <29, 29 - <33, 33 - <37, 37 - <41, 41 - <45.

The frequency distribution would be:

Percent of High School Dropouts Number of States

25 - <29 5

29 - <33 9

33 - <37 6

37 - <41 4

41 - <45 1

The histogram would be:


177

4. b) Multiply each percent by 242 to get the height of each bar.

5. a) 20.3% b) 24.9% c) 24.8% d) 30.0%

6. a) Home Health b) approx. 2.5% c) approx. 6.5% d) Hospital

7. a) ADN b) approx. 1%

12.1 16.2 25.9

187.8

0

50

100

150

200

Export Recycle Burn for Fuel Dump

Millions of Tires

Disposal of Tires in the U.S.

Unit 2 Topics in Algebra and Statistics Section 2.5 Measures of central tendency

179

Objectives for this section Calculate the mean for grouped or ungrouped data Calculate the median for data Calculate the mode for grouped or ungrouped data Interpret the effect on the mean, median or mode when data is modified Select which of the measures of central tendency is most appropriate for a given situation Interpret percentile rank Calculate quartiles using calculator

Summaries of Data

Data, like many people, seem to “hang out" at some favorite gathering spot. Numbers tend to be gregarious; they like each other’s company and tend to gather around some central point. Often we like to summarize a large amount of data with a single number that is somewhat representative of all of the data. In such a case, we use one of these gathering spots. Mean

Of such measures, one of the most widely used is the mean or average of the data. This is the measure most likely to be used by your instructor to assign you a grade at the end of the term. An instructor will often add up all of your test grades, divide by the number of tests you have taken to find your average for the term, and then assign you a grade based on this average. In other words, the instructor is using a single number (your test average) to represent several numbers (all of your different test grades). When the mean of an entire population is calculated, the symbol used for the mean is

(read this as “mu”). When the mean of a part of a population (called a sample) is

calculated, the symbol used for the mean is x (read this as “x bar”). In other words,

we would use to represent the average weight of all 18 year olds (the entire

population), but we would use x to represent the average weight of a representative

group (sample) of 18 year olds. We will use the symbol x to represent the mean whenever we calculate the mean using less than the entire population, which is usually the case.

To find the mean of a group of numbers, add all of the numbers together and then divide by how many numbers you have added together.


180

In mathematical notation, this process can be written:

Sample Mean: x

xn

and Population Mean: x

N

where x represents any one of the data, (read this as “sum of”)

indicates that all of the numbers should be added together, and n and N are how many numbers are added together.

Use x for the mean of a sample of a population and for the mean of the

entire population.

Example 1. Find the mean of the numbers 1 3 5 9 13 17.

Answer: Add all of the numbers together and divide by 6, which is how many numbers you added together:

1 3 5 9 13 17 48x

6 68.

Notice from this example that the mean does not have to be any one of the numbers that you added together.

The mean is not always the best representative for a set of data.

Example 2. Suppose that five persons have the following incomes:

$500,000 $20,000 $20,000 $10,000 $5,000

Find the mean and decide whether or not it is a good representative for the entire group.

Answer: Adding the incomes together and dividing by 5 gives

$500,000 $20,000 $20,000 $10,000 $5,000 $555,000

3 5$111,000

as the mean income of the group. This number obviously is not in any way representative of the bottom four incomes. The very large income at the top has raised the mean so much that it doesn’t really represent the incomes of most of the people in the group. In other words, the mean doesn’t necessarily represent the “average” person.


181

Example 3. Thirteen people were given a speeding ticket in a school zone. The

police officer reported to his supervisor that the mean speed was 32 mph for 12 tickets. What would be the mean speed for all 13 tickets if the 13th ticket was for 45 mph?

Answer:

12(32) 45

1333 mph .

Example 4. Sam bowled three games with scores of 210, 185, and 190. He needs to have a four-game average of 205 to qualify for an elite bowling league. What score does he have to bowl on the fourth game to have the average of 205 that he needs?

Answer: Let x = score in the 4th game.

210 185 190 x205

4

585 x205

4

585 x 820

x 235

Sam must bowl 235 or more to qualify for the league.

Median

Surveys often quote the median income rather than the mean income. The median is the middle number when the data are arranged in order of size. The median is sometimes a better representative of the data than the mean if the data contains a few numbers which are either very much smaller or very much larger than the rest of the numbers. The mean is greatly affected by such numbers.

Median is the middle number (or average of the two middle numbers) when data is arranged in order according to size.

Example 5. Using the data from Example 2, find the median.

Answer: The median is $20,000, since this is the middle income. Clearly, $20,000 is a much better indicator of the real income level of most of this group than is the mean which is $111,000.


182

If there is no middle number as in the following example, then the median is the average of the middle two numbers: Example 6. Find the median of the data: 8 10 6 1

Answer: First rearrange the numbers in numerical order from lowest to highest: 1 6 8 10

Then find the average of the two middle numbers, because there is not one middle number.

The median is

8 6

27 .

Note: You must always arrange the numbers in order of size when finding the median. If you have found the median correctly, there will be just as many numbers larger than the median as there are numbers smaller than the median. Example 7. Which would give you the better grade in a course--the mean of your

test scores or the median of your test scores? Answer: It depends upon what your test scores are.

Suppose your tests are 10, 85, and 100.

Your mean test score would be

10 85 100 195

653 3

.

Your median test score would be the one in the middle or 85. So in this case, the median is the better deal for you.

However, suppose that your test scores are 69, 71, and 100.

Your mean test score would be

69 71 100 240

803 3

.

Your median test score would be the one in the middle or 71. Here, the mean score is the better score.

Mode

Another measure for summarizing data is the mode.

Mode is the most frequently occurring number (or numbers) in a set of data. If each number occurs with the same frequency, then there is no mode.


183

Example 8. Find the mode for the following data: 9 9 7 5 5 5 4 2 2 1

Answer: The mode is 5, because 5 occurs three times, and no other number

occurs this many times. There may be more than one mode. Example 9. Find the mode for the data:

8 8 7 6 6 5 4 4 3 2 2 1

Answer: There are four modes 8, 6, 4, and 2, because each of them occurs twice, and no other number occurs more than once. For this reason, the mode is not used as much in statistical calculations as are the mean and the median.

There may not be a mode.

Example 10. Find the mode for the data. 8 8 2 2 5 5 4 4 6 6

Answer: There isn’t a mode, since each number occurs the same number of times (twice).

Situations do occur where the mode is the most appropriate quantity to use as may be seen from the following example. Example 11. Suppose there is a group consisting of the following ages,

15 19 19 19 19 19 19 19 19 25 31 31 31 31 31 31 31 31 35

and you wish to decide on the music to play for this group. Assume that you have surveys showing which tunes appeal to each age. Which measure, the mean or the median, would be better to use to determine the music to play for this group?

Answer: Neither. The mean and the median are both 25 years (compute them and see). If you played a mix of music tailored for this age, you would be appealing to perhaps only one person in this group of nineteen people! There are two modes for this group, 19 years and 31 years. You should probably play music suitable for each of these groups.


184

Example 12. Find the mean, the median, and the mode for the following set of data: 2 8 7 7 3 3 5 4 8 8

Answer: The mean is

2 8 7 7 3 3 5 4 8 8

105.5

To find the median, the data must be arranged in order either from smallest to

largest or from largest to smallest. Find the median as shown here:

Because there is not just one middle number, the median is the average

of the two middle numbers, which is

7 5

26 .

The mode is 8, since that number occurs most often (3 times).

Often, the mean, median and the mode are very close in value or even the same number. But this is not always the case as seen in the preceding example. If there is a mode, its value will be one of the data values. This is not true of either the median or the mean. Finding the mean, the median, and the mode when data is given using a frequency distribution or a histogram To begin let’s use the frequency distribution given below:

Height in Inches

Number of Persons of This Height

62 20

64 30

66 25

68 10

70 5

The average, or mean height, must take into account that there are a total of 90 persons, not just 5 persons. Also, it must take into account that each height occurs several times.


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The formula for the mean when data is given using a frequency distribution is

where x is a given number, f is how many times that number occurs, and f is how

many numbers there are in the group. Example 13. Find the mean, median, and mode of the data in the frequency

distribution above. Answer: The mean is calculated as follows:

total of all heights 62(20) 64(30) 66(25) 68(10) 70(5) 5,840x 64.9

total number of persons 20 25 30 10 5 90

The median for the group is either the middle number or the average of the two middle numbers. For this group, the median is the average of the 45th and the 46th heights from the bottom. Since both the 45th and the 46th numbers are in the 64-inch category, the median is 64 inches.

The mode is just the height which occurs most often, and is obviously also 64.

Example 14. Find the mean, the median and the mode for the incomes for a group whose incomes are distributed as follows:

Yearly Income Number of Persons with this Income

$80,000 40

$60,000 20

$40,000 30

$20,000 40

$10,000 20


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Answer: The mean is calculated as follows:

$80,000(40) $60,000(20) $40,000(30) $20,000(40) $10,000(20)x

40 20 30 40 20

$6,600,000

150$44,000 .

The median is the average of the 75th and the 76th numbers from the top. Both of these numbers are $40 000. Hence, the median is $40,000. There are two modes, $20,000 and $80,000, since these are the incomes which occur most often (40 times each).

Example 15. Find the mean income for the data given by the following frequency

distribution:

Classes Frequencies

0 - <10 20

10 - <20 10

20 - <30 30

30 - <40 60

40 - <50 30

50 - <60 10

Answer: Use the midpoint of each class (the average of the bottom of a class

and the bottom of the class just above) to represent that class. For instance, the data in the class 0 - <10 may be represented by the number 5, which is halfway between 0 (bottom of the class) and 10 (bottom of the class just above); all data in the class 10 - <20 may be

represented by the number 15

10 20 3015

2 2; and so on.

Sometimes this midpoint is called the “class mark.”


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Calculating

31.255(20) 15(10) 25(30) 45(30) 55(10) 5000

x20 10 30 60 30 10 160

Because we do not precisely know what numbers are in each class, this is certainly not an exact value for the mean. However, it is the best calculation we can make from the information given. The midpoint of each class is the best estimate of the actual numbers in the class. We make no attempt in this text to find the median or the mode when data is given in classes. Measures of Position

Measures of position are used to make comparisons, such as comparing the scores of individuals. Generally, measures of position are used when the data set is large. Two measures of position that we will discuss is percentiles and quartiles. Percentiles

There are 99 percentiles dividing a set of data into 100 equal parts. The percentile rank of a number is the percent of the data which is below the number. For instance, if 80% of the group ranks below an individual, then that individual would have a percentile rank of 80. The correct statement is that a score is the 80th percentile, not the 80% percentile. That is, don’t use % together with the word percentile. If a person scored in the 80th percentile on a standardized test, this means that 80% of the people taking the test had a score that was below that person’s score. In other words, that person outperformed 80% of the people taking the test. Example 16. Ann Brown took a Mathematics placement test. Her score was in the

75th percentile. Explain what that means. Answer: This means that Ann Brown outperformed 75% of the people taking the

Mathematics placement test. Seventy-five percent, 75%, of the persons taking the test scored below Ann.

Example 17. On a standardized test taken by 5,000 students, you scored in the 70th

percentile. How many students scored below you? How many scored above?

Answer: 5000(0.70) = 3500 scored below you.

5000 – 3500 – 1 (you) = 1499 scored above you.


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Quartiles

Quartiles divide data into four groups rather than one hundred groups as do percentiles. The first quartile (sometimes called the lower quartile) is the number with 25% of the data below it; the second quartile is the number with 50% of the data below it; and the third quartile is the number with 75% of the data below it. Finding Quartiles using the TI-30XS MultiView Calculator

1. Press (twice)

2. Arrow down to highlight 4: Clear All

3. Press This will clear out any data that is left over from a previous problem.

4. Enter data in L1

5. If the frequency of the data is more than

one (grouped data), enter frequency in L2.

Each column should have the same number

of entries.

6. Press

7. 1: 1-Var Stats will be highlighted

Press

8. Highlight L1 in the Data row

Press

9. If you entered frequencies in L2, highlight

L2 in FRQ row.


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10. If you did not enter any frequencies in L2, then ONE should be highlighted.

11. Arrow down to highlight CALC

12. Press

13. Arrow down to 8: for Q1 9: Med for Q2 and A: for Q3

Example 18. An electronics store is concerned about high turnover of its sales staff.

A survey was done to determine how long (in months) their sales staff had been in their current positions. The responses of 27 sales staff follow. Determine Q1, Q2, and Q3.

25, 3, 7, 15, 31, 36, 17, 21, 2, 11, 42, 16, 23, 19, 21, 9, 20, 5, 8, 12, 27, 14, 39, 24, 18, 6, 10

Answer (without using the calculator):

Step 1: Order the data as follows:

2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 21, 23, 24, 25, 27, 31, 36, 39, 42

Step 2: Find the median, Q2.

The fourteenth number in the list, 17, is the median since there are 13 numbers before it and 13 numbers after it. Step 3: Find the median of the lower half of the data, Q1.

The lower half consists of 13 numbers; therefore, the 7th number of the data set is the median of the lower half. The 7th number is 9. Step 4: Find the median of the upper half of the data, Q3.

The upper half consists of 13 numbers. These numbers are the 15th through 27th numbers in the list. The 21st number is the median of the upper half. The 21st number is 24. Therefore, Q1 is 9, Q2 is 17, and Q3 is 24.

Or you can use the calculator as outlined above.


190

1. Find the mean, the median, and the mode of the following data:

7 7 7 7 7

2. Find the mean, the median, and the mode of the following data:

8 2 1 4 8 4 1 3 1 8

3. Find the mean, the median, and the mode of the data given by this frequency distribution:

Income Number of Families Having this Income

$18,000 7

$20,000 8

$24,000 7

$28,000 15

$34,000 7

$85,000 1

4. Make up a set of five scores (not all the same) in which the mode, median and

mean are all the same. 5. The mean score of 30 tests is 72. What is the sum of the 30 test scores?

6. The mean of a set of 18 scores is 83. Two more students take the test and score 65 and 70. What is the new mean?

7. If the mean salary of one department of six employees is $24,000, and the

mean salary of another department of fourteen employees is $16,000, what is the mean salary of both departments combined?

8. To receive an A in Math 155, Libby must have a mean of 90 or better on four

exams. Libby’s first three exam scores are 77, 92 and 96. What grade does she need on the fourth exam to receive an A in Math 155?


191

9. A group of ten persons consists of one 20-year-old person and nine 40- year-old persons.

a) What is the average age for the group? b) What is the median age for the group? c) What is the mode for the ages of the group?

10. Another group consists of one 40-year-old person and nine 20-year-old persons.

a) What is the average age of the group? b) What is the median age of the group? c) What is the mode for the group age?

11. If a group of five persons whose average age is 20 is combined with a group of 25 persons whose average age is 40, what is the average age of the resulting group? Round your answer to the nearest tenth.

12. a) The average age of 10 persons in an office is 30. What will the

average age become if a 52-year-old person is added to the staff?

b) The average of 100 persons in an office is 30. What will the average age become if a 52-year-old person is added to the staff? Round your answer to the nearest tenth.

13. A group of 20 persons with an average weight of 200 pounds is combined with a

group of 60 persons with an average weight of 120 pounds. What is the average weight of the resulting group?

14. a) If every person in a group gains 5 pounds, how much does the average

weight of the group change?

b) If the hourly wage of every person in a group doubles, what happens to the average hourly wage for the group?

15. The median weight of a group of 43 athletes is 180 pounds. If one person

weighing 320 pounds and another person weighing 175 pounds both join the group, find the median weight of the resulting group.


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16. Find the mean, the median, and the mode of each of the following groups of data: a) 15, 13, 10, 7, 6, 9, 10

b) 3, 5, 8, 13, 21

c) 0, 1, 1, 2, 3, 4, 16, 16

d) In a comparative study of stereo headphones, 12 different models were purchased. The prices, in dollars, were as follows: 37, 50, 60, 69, 50, 75, 40, 45, 70, 60, 30, 50.

e) In a second comparative study of stereo headphones, 12 different models were purchased. The prices, in dollars, of this second group were as follows: 60, 30, 50, 25, 28, 30, 30, 35, 39, 40, 75, 29

f) The number of days that you must wait for a marriage license in various states in the U.S. is given by the chart.

g) Sam had the following bowling scores in league on Saturday: 175, 150, 160, 180, 160, 183, 287

h) Tom had the following bowling scores in league on Saturday: 180, 130, 161, 185, 163, 185, 186

i) 3, 5, 5, 8, 9

j) 4, 10, 9, 8, 9, 4, 5

k) 6, 5, 4, 7, 1, 9


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l) The table to the right gives the test scores on a math test.

m) The Don Shoe Co. has a total of 25 employees with the following classes as salaries as shown in the table. (Find the mean only.)

Salary Frequency

$20,000 - <$25,000 6

$25,000 - <$30,000 8

$30,000 - <$35,000 5

$35,000 - <$40,000 4

$40,000 - <$45,000 2

17. Make a histogram for each of the tables in the previous problem.

18. The following statement appeared in a company newsletter: “I explained that the average life of an incandescent bulb was 720 hours. This number is calculated by bulb manufacturers by turning on several hundred light bulbs and leaving them on until half have burned out. This is quoted as the 'average bulb life.” What is the correct statistical name for the 720 hours being given as the average life of a bulb?

19. When a national sample of heights of kindergarten children was taken, Evan

was told that he was in the 40th percentile. Explain what that means. 20. Out of 200 people in Donna’s graduating class, Donna’s percentile rank was 72.

How many classmates ranked below Donna? How many ranked above?


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21. The cost, in cents, per half-cup serving of the top 20 rated brands of vanilla ice cream, as reported in July 2002 issue of Consumer Reports, are as follows:

17, 21, 27, 27, 28, 33, 80, 17, 24, 27, 28, 28, 38, 81, 20, 25, 27, 28, 31, 74

a) Determine Q1, Q2, and Q3.

b) What price has a percentile rank of 75?

22. Find the quartiles of the following grouped data set:

Income Number of Families Having this Income

$18,000 7

$20,000 8

$24,000 7

$28,000 15

$34,000 7

$85,000 1

Unit 2 Topics in Algebra and Statistics Section 2.5 Answers to selected Problems

195

1. Mean, median and mode are each

7. 2. Arrange in order

1, 1, 1, 2, 3, 4, 4, 8, 8, 8 Mean = 4 Median = 3.5 Mode = 1 and 8

3. $26,600 = mean

$28,000 = median and mode

4. The numbers 1 3 3 3 5 have a mean of 3, a median of 3, and a mode of 3.

5. sum

7230

; hence, sum =

30(72) = 2160.

6. Sum of the 18 scores is 18(83) or 1494. The sum of all 20 scores is 1494 + 65 + 70 or 1629. Thus, the

mean is 1629

81.4520

.

7. $18,400

8.

77 92 96 x

904

x = 95

9. a) 38 b) 40 c) 40

10. a) 22 b) 20 c) 20

11. 36.7 (to the nearest tenth)

12. a) 32 b) 30.2 (nearest tenth)

13. 140 pounds

14. a) increases by 5 pounds

b) doubles

15. There is no change in the median since 180 pounds is still in the middle of the weights. One of the new persons weighed more than 180 and the other new person weighed less than 180.

16. Mean median mode

a) 10 10 10 b) 10 8 none c) 5.375 2.5 1, 16 d) 53 50 50 e) 39.2 32.5 30 f) 1.58 0.5 0 g) 185 175 160 h) 170 180 185 i) 6 5 5 j) 7 8 4, 9 k) 5.33 5.5 none l) 68 70 70 m) $30,100

19. 40% of the kindergarten children who participated in the survey are shorter than Evan.

20. a) 144 below

b) 55 above

21. a) Q1 = 24.5 cents Q2 = 27.5 cents Q3 = 32 cents

b) 33 cents

22. Q1 = $20,000 Q2 = $28,000 Q3 = $28,000

Unit 2 Topics in Algebra and Statistics Section 2.6 Measures of dispersion

197

Objectives for this section Calculate the range for data Calculate the standard deviation for data Interpret the change in the range and standard deviation if data is modified Make inferences about the size of the standard deviation from a histogram

Calculating the Range and the Standard Deviation

The mean, the median, and the mode are helpful, but not completely adequate, to describe data. For instance: The mean of the numbers 0 and 100,000 is 50,000.

The mean of the numbers 49,999 and 50,001 is also 50,000.

Here we have two sets of data with the same mean but with completely different characteristics. The first data are two widely spaced numbers, while the second data are two very closely spaced numbers. Range

One measure used to distinguish between very closely spaced numbers and very widely spaced numbers is the range. The range is the difference between the largest number in a group and the smallest number in the group.

Range = Largest Number - Smallest number

The range of the data consisting of 0 and 100,000 would be 100,000, since 100,000 - 0 =100,000. The range of the data consisting of 49,999 and 50,001 would be 2, since 50,001 – 49,999 = 2. Example 1. Find the range of the data: 7 12 4 5 7 9 17 8 8.

Answer: The range is 13 since, largest - smallest = 17 - 4 = 13.

Note that the range uses only the smallest and the highest number in the data. Thus, the data 1 1 1 1 9 9 9 9 and the data 1 5 5 5 5 5 5 9 could have the same range (and the same mean). In each case, the range would be 8, and the mean would be 5. You should verify this.


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Standard Deviation

Clearly, there is a big difference between the two sets of data given in the preceding paragraph. The first set of data consists of several small numbers and several large numbers with nothing in between, whereas the second set of data consists mostly of middle size numbers which are all fairly close to the mean. There is a measure of the dispersion of numbers which is based on their distances from the mean. It is called the standard deviation. Calculating the standard deviation for a sample of a larger population involves performing the following steps: 1. Find the mean of the numbers.

2. Subtract the mean from each one of the numbers. Square each of the resulting numbers.

3. Add together all of the numbers you calculated in step 2.

4. If your data is a sample, divide by one less than the number of data you have. If your data represents the entire population, divide by the number of data. The result you have now obtained is called the variance.

5. Take the square root of the result. This is the standard deviation.

Standard deviation (for a sample):

2

x xs

n 1. Unless

otherwise instructed, we will use this version of the standard deviation throughout this section assuming that the data given is a sample of the

population. The symbol (read as “sum”) means to add together

the numbers that follow it.

The other form of the standard deviation is written " " and is used when the data

given is the entire population. Its formula is

2

x

N. We will generally

not use this form of the standard deviation in this section, since we generally consider the small data sets found in the examples and problems to be samples of a larger population.


199

The following examples are given to show that the standard deviation can be calculated using the previously mentioned steps. Example 2. Find the standard deviation for the following data:

2, 5, 9, 12

Answer: Adding the numbers and dividing by 4 yields a mean of 7.

Then calculate

2 2 2 22 7 5 7 9 7 12 7

3 giving 4.40 for

the standard deviation when rounded to two decimal places. Example 3. What happens to the standard deviation if we add 5 to each of the

numbers in the previous example? Answer: Try it and see. The numbers become 7, 10, 14, 17. The mean is now 12.

The standard deviation is now

2 2 2 2

7 12 10 12 14 12 17 12

3= 4.40 when rounded to

two decimal places. Thus, there is no change in the standard deviation when each number in a set of data is increased by the same amount. In other words, the numbers are no more spread out after 5 is added to each one than they were before 5 was added.

Standard Deviation for Grouped Data (data given using a frequency distribution):

2

f

x-x fs=

-1

where x is an individual number, f is how many times this number occurs, f is how

many numbers there are altogether, and x is the mean for the data. When data are given by classes, x is taken to be the midpoint of the class.


200

The scientific calculator recommended for this course will calculate the sample standard deviation or the population standard deviation for regular data sets or

grouped data sets. They appear on the screen as Sx and x , respectively. For this course, you should use your calculator to find the standard deviation of a data set. Calculating the mean and the standard deviation using the TI-30XS MultiView Calculator

1. Press (twice)

2. Arrow down to highlight 4: Clear All

3. Press

This will clear out any data that is left over from a previous problem.

4. Enter data in L1

5. If the frequency of the data is more than one (grouped data), enter frequency in L2. Each column should have the same number of entries.

6. Press

7. 1: 1-Var Stats will be highlighted.

Press

8. Highlight L1 in the DATA: row.

Press

9. If you entered frequencies in L2, highlight L2 in FRQ: row.


201

10. If you did not enter any frequencies in L2, then ONE should be highlighted.

11. Arrow down to highlight CALC.

12. Press

13. Arrow down to 2: x for the mean of the data. 3: Sx is the sample standard deviation. 4: σx is the population standard deviation. 9: Med is the median.

The following examples using frequency histograms illustrate the concepts of mean, median, mode, range, and standard deviation when data is given using a histogram or a frequency distribution: Use the frequency histogram below for the next five examples:

Corresponding frequency distribution:

Height Number of Persons of a Given Height

62 30

64 25

66 20

68 10

70 5

Example 4. Find the mode for the heights.

Answer: 62 inches, because that height occurs the most times (there are 30 people with this height). It is the category with the tallest bar.

0

10

20

30

40

62 64 66 68 70Height in inches

Number of Persons of a Given Height


202

Example 5. Find the median for all of the heights.

Answer: Without using the calculator: Since there is an even number of heights (90), the median is the average of the two middle heights. From the preceding table, the 45th height from the bottom is 64, and the 46th height from the bottom is also 64. Hence, the median height is 64 inches, which is the average of these two heights. Note that the middle height in the table is not necessarily the median. The median is the middle number when the data are arranged in order.

Using the calculator: Med = 64 inches

Example 6. Find the mean of all of the heights.

Answer: Without using the calculator:

62(30) 64(25) 66(20) 68(10) 70(5) 5,810x

30 25 20 10 5 9064.6 inches

You must add up all of the heights, and divide by the number of persons. The height 62 inches occurs 30 times; the height 64 occurs 25 times; etc. The total number of persons is the sum of the numbers in each of the categories, which is 90.

Using the calculator: x 64.6 inches

Example 7. What is the range of all of the heights?

Answer: Subtract the smallest height from the largest height. 70 - 62 = 8 inches

Example 8. What is the standard deviation of all of the heights?

Answer: Using the calculator: 2.4 inches

Note that the standard deviation has the same units as the mean. In this case, both are measured in inches.


203

Example 9. The following data sets given by frequency histograms have progressively larger standard deviations. That is, a), b), c), d), e), and f) progress from smallest possible standard deviation to largest possible standard deviation for data consisting of thirty 0's, 1’s, 2’s, 3’s, and 4’s.

a) b)

c) d)

e) f)

A small standard deviation tells us that most of the data are close to the mean, and a large standard deviation tells us that most of the data are far away from the mean. A set of data with a large standard deviation is said to be more variable than a set of data with a small standard deviation.

0

5

10

15

20

25

30

35

0 1 2 3 40

5

10

15

20

25

30

0 1 2 3 4

0

2

4

6

8

10

12

0 1 2 3 4

0

1

2

3

4

5

6

7

0 1 2 3 4

0

2

4

6

8

10

0 1 2 3 4

0

2

4

6

8

10

12

14

16

0 1 2 3 4


204

1. Calculate the range and the standard deviation for each of the following:

a) 18, 22, 20, 17, 21

b) 500, 4, 8, 96, 55, 3, 1

c) 6, 6, 6, 6, 6, 6, 6, 6

d) 85, 88, 86, 80, 32

e) 96, 93, 90, 88, 91

2. The final grades for two Math 155 instructors have been analyzed. Instructor A’s class had a GPA of 3.0 and a standard deviation of 0.5, while Instructor B’s class had a GPA of 3.0 and a standard deviation of 1. Which Instructor’s class would you prefer to be in and why?

3. In a hospital, each patient’s pulse rate is taken three times per day. Patient

A’s rates were 73, 75 and 76, while Patient B’s rates were 74, 87 and 63.

For each patient’s pulse rate find:

a) mean b) range c) standard deviation

4. The ages of a group of children are distributed as shown:

Age Number of Children of this Age

5 40

10 20

15 10

a) Find the mode for the ages.

b) Find the median age.

c) Find the mean age.

d) Find the standard deviation for the ages.


205

5. The incomes for several families are distributed as shown:

Family Income in Dollars Number of Families

0 - <10,000 10

10,000 - <20,000 15

20,000 - <30,000 5

a) What is the mean family income?

b) What is the standard deviation for the family incomes?

6. Find the mean and the standard deviation from the table below:

Hours of Television Watched per Week

HOURS FREQUENCY

1 - <3 8

3 - <5 19

5 - <7 27

7 - <9 6

7. Use the following histograms to answer the questions:

Example 1

0

1

2

3

4

1 2 3 4 5

Example 2

0

2

4

6

8

1 2 3 4 5

a) What number is the mean, mode and median for each histogram?

b) Which is less variable (has a smaller standard deviation)?

c) What is the range for each?


206

8. Find the range and standard deviation for the following data sets:

a) 15, 13, 10, 7, 6, 9, 10

b) 3, 5, 8, 13, 21

c) 0, 1, 1, 2, 3, 4, 16, 16

d) In a comparative study of stereo headphones, 12 different models were purchased. The prices, in dollars, were as follows: 37, 50, 60, 69, 50, 75, 40, 45, 70, 60, 30, 50.

e) In a second comparative study of stereo headphones, 12 different models were purchased. The prices, in dollars, of this second group were as follows: 60, 30, 50, 25, 28, 30, 30, 35, 39, 40, 75, 29.

f) The number of days that you must wait for a marriage license in various

states in the U.S. is given by the chart.

g) Sam had the following bowling scores in league on Saturday: 175, 150, 160, 180, 160, 183, 287.

h) Tom had the following bowling scores in league on Saturday: 180, 130, 161, 185, 163, 185, 186.

i) 3, 5, 5, 8, 9

j) 4, 10, 9, 8, 9, 4, 5

k) 6, 5, 4, 7, 1, 9


207

l) The table to the right gives the test scores on a math test.

m) The Don Shoe Co. has a total of 20 employees and only 4 different salaries as shown in the table.

Salary Frequency

$25,000 8

$28,000 6

$30,000 4

$45,000 2

Writing exercises

9. Suppose that each number in a set of data is increased by 10.

a) What effect does this have on the mean for the data? b) What effect does this have on the median for the data? c) What effect does this have on the mode for the data? d) What effect does this have on the range of the data? e) What effect does this have on the standard deviation for the data?

10. Explain in your own words why the range may not be a good measure of dispersion.

11. If all the numbers in a set of data are equal, what is the range? What is the

standard deviation? What is the mean?


208

1. a) Range = 5 x 19.6 s = 2.07

b) Range = 499 x 95.3 s = 182

c) Range = 0 x 6 s = 0

d) Range = 56 x 74.2 s = 23.8

e) Range = 8 x 91.6 s = 3.05

2. answers vary

3. a) 74.7 for A 74.7 for B b) 3 for A 24 for B c) 1.5 for A 12.0 for B

4. a) 5 b) 5 c) 7.9 d) 3.7

5. a) $13,333 b) $6989

6. x 5.03 s = 1.7

7. a) 3 b) Example 2 c) 4

8. range standard deviation a) 9 3.2 b) 18 7.2 c) 16 6.7 d) 45 14.1 e) 50 15.2 f) 5 1.7 g) 137 46.6 h) 56 20.6 i) 6 2.4 j) 6 2.6 k) 8 2.7 l) 50 11.9 m) $20,000 $5848

9. a) adds 10 b) adds 10 c) adds 10 d) none e) none

10. Calculation of the range involves only the largest number and the smallest number. The other data are ignored.

11. range = 0 standard deviation = 0 Every number is the mean.

Unit 2 Topics in Algebra and Statistics Section 2.7 Unit review

209

Important Terms and Formulas

Frequency: How many times a given number in a group of data occurs.

Frequency Distribution: Chart with classes in first column and amount of data in

each class (frequency) in second column. A frequency distribution must be done in order to create a frequency histogram.

Frequency Histogram: Vertical bar graph with classes on the horizontal (x) axis

and amount of data in each class (frequency) on the vertical (y) axis. Classes must all be the same width.

Time-Series Plot: Uses data that is measured at different points in time.

Obtained by plotting the time in which a variable is measured on the horizontal axis and the corresponding value of the variable on the vertical axis.

Circle Graph: Good way to show what fraction of data is in each class.

The size of the slice shows how much data is in that class.

Mean: Average of data. If the data is given only by classes,

compute the midpoint of each class and use it to represent all of the data in the class. If data occur more than one time, multiply each number by its frequency, add the results together, and then divide by the total of all the frequencies.

Median: Middle number in a group of data when data is arranged

in order from smallest to largest. If there is no middle number, the median is the average of the two middle numbers.

Mode: Number occurring most frequently in a group of data. If

each number occurs just once, there is no mode. There can be several modes.

Unit 2 Topics in Algebra and Statistics Section 2.7 Unit review

210

Range: The difference between the largest and the smallest member of data.

Standard Deviation: A measure of how far data are away from the mean. A

small standard deviation indicates that most data are close to the mean.

Percentile: The percentile rank of a number in a group of data is the

percent of the data which are less than the given number. Generally, the decimal portion and the percent symbol (%) are both dropped.

Quartile: The quartile rank of a number in a group of data is the

number of quarters of the data which are below that number. The third quartile is above exactly three quarters of the data. The first quartile is above the bottom quarter of the data.

THE FOLLOWING INFORMATION AND NO OTHER WILL BE FURNISHED ON THE UNIT 2 TEST.

Equation of a Line y=mx b where m is the slope of the

line, and b is determined by a point on the line. The point (0, b) is the y-intercept of the line.

Slope of a Line

2 1

2 1


run horizontal change x x

amount y changes when x increases by 1

Mean x

xn

x

N

Unit 2 Topics in Algebra and Statistics Section 2.7 Unit review problems

211

1. Give the coordinates (whole numbers only) and the quadrants for each of the

points shown on the axes below:

2. a) Give the coordinates of a point which is two units to the left of the y- axis and three units above the x-axis. b) In which quadrant is the point in the previous question located?

c) Find the coordinates of a point lying on the y-axis and 5 units below the x-axis.

d) In what quadrant is the point in the previous question located?

3. a) Graph y = 2.

b) Graph x = 1.

c) Graph 2x - y = 4.

d) Graph y = 3 - 2x.


212

4. a) If the slope of a line is -4 and x increases by 8, find the amount by which

y changes.

b) Find the equation of the vertical line through (8, -5).

c) Find the equation of the horizontal line through (0, 2).

d) Find the x-intercept of the line 5x + 3y = 30.

e) Find the y-intercept of the line x = 17.

f) Find the slope of the line through (2, 7) and (9, 7)

g) Find the equation of the line in the previous problem.

h) What is the highest power which may appear on each of the variables in the equation of a line?

i) Draw an example of a line with a positive slope.

j) Draw an example of a line with a zero slope.

k) Draw an example of a line with a negative slope.

l) Draw an example of a line with an undefined slope.

m) Find the slope of the line through (2, 6) and (-1, 24).

n) Find the equation of the line in the previous question.

5. The height of a tree is given by the equation y = 2x + 1, where x is the number of years since the tree was planted and y is the height of the tree measured in meters. a) Find the amount the tree grows each year.

b) Find the height of the tree when it was planted.

c) Find the height of the tree 5 years after it is planted.

d) Graph the equation.


213

6. A machine is purchased for $12,000 and is sold for scrap 8 years later for

$2000. Its value decreases according to a straight line graph.

a) Find the equation of the line showing the relationship between y, the value of the machine, and x, the number of years since the machine was purchased.

b) Calculate the number of dollars the value of the machine decreases each

year.

c) Calculate the value of the machine 5 years after it was purchased.

7. a) Find the grade of the road shown below and express it as a percent.

b) Find the pitch of the roof shown below and write it as a fraction with a denominator of 12.

8. Find the variable cost rate for Jane Doe Company:

Jane Doe Company

Units Sold

Manufacturing Expense


Low activity level 20,000 $50,000


214

9. a) Graph y = 2x - 2 using the axes to the right.

b) Graph y = 4 using the axes to the right.

c) Graph x + 2y = 6 using the axes to the right.


215

d) Graph x = -3 using the axes the right.

10. Use this data to answer the questions which follow: 30, 15, 27, 30, 15, 21

a) Find the mean.

b) Find the median.

c) Find the mode.

d) Find the range.

e) Find the standard deviation.

11. Use this frequency distribution to answer the questions which follow:

Score Number of Persons Making this Score

40 5

60 4

80 3

100 8

a) Find the median. b) Find the mode.

c) Find the mean (rounded to one decimal place). d) Find the range.

e) Find the standard deviation (rounded to one decimal place).


216

12. Construct a frequency distribution table so that the first class beginsat 1 and the class interval is 5.

14, 14, 18, 20, 13, 5, 19, 20, 20, 20, 19, 23, 19, 3, 3, 5, 5, 5, 11, 12, 12, 11, 15, 22, 23, 22, 22, 24, 7, 9, 9, 6, 16, 15, 16, 17, 18, 18, 9, 9, 7, 8, 19, 6 13. a) If each number in a group of numbers is multiplied by 3, what is the effect

on the mean?

b) If each number in a group of numbers is multiplied by 3, what is the effect on the standard deviation?

c) If each number in a group of numbers is increased by 3, what is the effect

on the mean?

d) If each number in a group of numbers is increased by 3, what is the effect on the standard deviation?

14. a) If your percentile rank in a group of 800 persons is the 40th percentile

how many of the group rank higher than you?

b) What percentile rank is usually closest to the median?

c) What is the highest possible percentile rank a number may have? Explain.

d) What is the lowest possible percentile rank a number may have? Explain.

15. On a standardized test taken by 1000 students, you scored in the 60th

percentile.

a) How many students scored below you?

b) How many scored above you?

16. Determine the quartiles for the grouped data:


217

1. a) (-5, 4) II b) (0, -3) None c) (6, 2) I d) (-1, -5) III e) (-3, 0) None

2. a) (-2, 3) b) II c) (0, -5) d) None

3. a)

b)

c)

d)

4. a) changes by -32

b) x = 8 c) y = 2 d) (6, 0) e) None f) 0 g) y = 7 h) 1

i) – l) m) -6 n) y = -6x + 18

5. a) 2 m b) 1 m c) 11 m

6. a) y = -1250x + 12,000 b) $1250 c) $5750

7. a) 40

0.08 8%500

b) Normally pitch is given as a fraction with a denominator of 12. So write the slope in the form:

6 x

18 12

72 = 18x x = 4

The pitch of the roof is 4

12.

8. $10/unit sold


218

9. a)

b)

c)


219

d)

10. a) 23 b) 24 c) 30 and 15 d) 15 e) 7.0

11. a) 80 b) 100 c) 74 d) 60 e) 25.2

12.

Category Number in Category

1 - <6 6

6 - <11 9

11 - <16 9

16 - <21 14

21 - <26 6

13. a) multiplied by 3 b) multiplied by 3 c) increased by 3 d) unchanged

14. a) 479 b) 50th c) 99 100% cannot be below the

number since the number is part of the group.

d) 0 The lowest number has this rank.

15. a) 600

b) 399

16. Q1 = 60 Q2 = 70 Q3 = 80

Unit 2 Topics in Algebra and Statistics

221

UNIT 3

APPLICATIONS OF PERCENT AND ALGEBRA

Unit 3 Applications of Percent and Algebra Section 3.1 Proportions

223

Objectives for this section Write stated information as a ratio Solve proportions Solve applied problems using proportions

Ratios

Ratio is just another word for fraction. The ratio 7 to 10 may be written as the

common fraction 7

10 , as the common fraction 7 /10, as the ratio 7:10, or even as the

decimal fraction 0.7. The order in which the ratio is stated is important. For instance, if you have 4 nickels and 7 dimes in your pocket, the ratio of nickels to

dimes is 4 to 7 or 4

7, but the ratio of dimes to nickels is 7 to 4 or

7

4. The numerator

or the denominator of a ratio may not be a fraction or a decimal. They must both be whole numbers. Example 1. Suppose a group consists of 20 women, 30 men, and 40 children.

a) Find the ratio of children to women.

b) Find the ratio of women to the total number in the group.

Answers: a) 40 to 20, or 2 to 1, when the fraction is reduced to lowest terms. The answer may also be written 2:1.

b) 20 to 90, or 2 to 9, when the fraction is reduced. The answer may also be written 2:9.

Thus, the quantity mentioned first in a ratio is normally the numerator (top), and the quantity mentioned last is usually the denominator (bottom) of the ratio. The batting average for a baseball player is the ratio of the number of hits to the number of times at bat expressed as a decimal rounded to three decimal places. Example 2. What is the batting average for a baseball player who gets 13 hits in 40

times at bat?

Answer: 13

400.325

Sometimes the answer is written 325 without the decimal in front of the 3. Sports broadcasters say, “The player is batting 325” without mentioning the decimal.


224

Proportions

A proportion is the equation created when two fractions (or ratios) are set equal to each other. There can be only one fraction on the right side of the equal sign and one fraction on

the left side of the equal sign. Thus, the equation x 5 3x 1

2 7 5 3 is not considered to

be a proportion, since there is more than one fraction on each side of the equal sign. Examples of proportions stated in different ways:

Example 3. 3 7

x 10

Example 4. x over 5 equals 7 over 15. (meaning x 7

5 15)

Example 5. The ratio of 7 to 9 is the same as the ratio of 15 to x.

(meaning 7 15

9 x)

Example 6. 2:3 = 10:15. (meaning 2 10

3 15)

Example 7. 4/5 = x/20. (meaning 4 x

5 20)

Example 8. 5 is to 10 as 4 is to 8. (meaning 5 4

10 8)

Example 9. 1:2 :: 50:100. (This is an old notation meaning 1 50

2 100)

Notice that there are several ways to write proportions. We will generally write them as shown in Example 3. Also, notice in the following examples that whenever two fractions are equal, cross multiplying yields two numbers which are equal.


225

Example 10. 3 6

5 10. Therefore, (3)(10) = (5)(6), or 30 = 30

Example 11. 7 70

3 30. Therefore, (7)(30) = (3)(70), or 210 = 210

When the proportion contains an unknown, we solve it as follows:

Example 12. Solve x 1

8 2

Answer: Cross multiplying yields: 2x = 8(1), or 2x = 8. Hence, x = 4 is the solution.

Example 13. Solve 5 2

x 7

Answer: Cross multiplying yields: 5(7) = 2x, or 35 = 2x.

Hence, 35

x or 17.52

.

Example 14. Solve a c

b d for the letter b.

Answer: ad = bc Cross multiply

ad

bc

Divide both sides by c

Example 15. Solve a c

b d for the letter d.

Answer: ad = bc Cross multiply

bc

da

Divide both sides by a


226

Applications of Proportions

Hair Design:

In the study of Cosmetology, students learn from the Principles of Design, “Proportion is the direct correlation of size, distance, amount and ratio between the individual characteristics when compared with the whole. A design with proper proportion is arranged to be harmonious and graceful. Dividing components into thirds using the Golden Ratio of divine proportion (3 to 2, or 2 to 3) balances the hair design in relation to the entire image (not just the head) and reinforces the artistic principle of proportion. . . To the professional of hair design, the Golden Ratio is the key to a well defined hairstyle.”

This hair design is well balanced. The face is two thirds of the total area.

This hair design is out of proportion. Her face is 3/4 of the total area, but her hair is only 1/4 of the total area.

Example 16: Is this hair design well balanced?

Answer: The amount of face is approximately 1/3 of the total area. Her hair equals 2/3 of the total area. This creates the perfect balance for her features. Therefore, it is proportional.


227

Solutions: The strength of a solution (such as water containing dissolved salt or

water containing alcohol) is the ratio of the amount of solute (the substance dissolved) to the total amount of solution. So if we have two solutions with the same strength, then the ratio of solute to solution for one must be the same as the ratio of solute to solution for the other.

Example 17. If 300.0 milliliters of a solution contains 10.0 grams of salt, how much

salt must 800.0 milliliters of solution contain if it is to be the same strength as the original solution? Round your answer to the nearest tenth.

Answer: Let x = the unknown amount of salt in grams.

Write the proportion 10 grams x grams

300 milliliters 800 milliliters making sure that

you keep the same units on the top on both sides and the same units on the bottom on both sides.

Solving yields 300 x = 8000

or 8000

x300

26.7 grams .

Example 18. If 200 mL of a solution can be made using 30 mL of alcohol, how much solution of the same strength can be made using 75 mL of alcohol?

Answer: Let x = the unknown amount of solution in mL.

Write the proportion 30 mL of alcohol 75 mL of alcohol

200 mL of solution x mL of solution.

making sure that you put the same units on the top on both sides and

the same units on the bottom on both sides.

Solving by cross multiplying yields

30 x = 15,000 or x = 500 mL


228

Scales on maps: Generally a distance scale is given on any map, such as 1 inch = 20 miles. If you have distances on a map and wish to find actual distances, or if you have actual distances and wish to find the distance on the map, you can use a proportion to solve the problem.

Example 19. Suppose 1.0 inch = 40 miles. How many inches on the map will

represent 100 miles? Answer: Let x = the unknown amount of inches.

Write the proportion 1 inch x inches

40 miles 100 miles.

Solve 40x = 100 or x = 2.5 inches

Be sure to include the units (inches, milliliters, etc.) when solving applied problems. Enlargements: When a picture is enlarged, the ratio of the width to the length

must be maintained or else the enlargement will leave out part of the picture. Generally, the latter is what happens, since different sizes of photographic paper have different width-to-length ratios.

Example 20. Does 5 x 7 photographic paper have the same width-to-length ratio as

11 x 14 photographic paper?

(In the United States, paper dimensions are generally given in inches, so 5 x 7 means 5 inches wide and 7 inches long.)

Answer: This is the same as asking “Does 5 11

7 14?”

Cross multiplying gives the result 77 = 70, which obviously is not true. Hence, an enlargement of a 5 x 7 photo will not fit exactly on 11 x 14 paper, and some of the picture will be lost.

Example 21. Two more standard U.S. sizes of photographic paper are 8 x 10 and 16

x 20. Can an 8 x 10 photo be enlarged so that it exactly fits a sheet of 16 x 20 paper?


229

Answer: Yes. When cross multiplied, 8 16

10 20 yields 160 = 160, which is a true

result. Example 22. If an 11 x 14 photo is enlarged so that the longer dimension of the

enlargement is 49 inches, what must be the length of the shorter dimension of the enlargement if the fit is to be perfect with nothing left out?

Answer: Let x = the unknown amount of inches.

Write the proportion 11 x

14 49.

Cross multiply giving the equation 14x = 11(49).

Solving for x and rounding to the nearest whole inch, x = 39 inches.

Example 23. Why don’t movies fit perfectly on TV screens?

Answer: The ratio of the width to the height of a movie screen is much greater than the ratio of the width to the height of a TV screen, because movies are usually shown on a wide screen, and TV screens can be almost square. There are at least three choices on how to make a movie fit a television screen:

a) Enlarge so that the film fits top-to-bottom and allow some of the

movie to be lost off the sides.

b) Enlarge the movie so that it fits side-to-side and insert black bands above and below the picture to fill in the blank area of the TV screen.

c) Distort the picture so that the movie fits exactly on the television

screen. This creates the very tall, skinny characters you may have seen when a movie which was originally shown on a very wide screen is shown on TV.

The usual ratio of the width to the height on a TV screen is 4 to 3. The ratio on some newer television sets is 16 to 9, which is much closer to the ratio of the width to the height on a movie screen.

Unit 3 Applications of Percent and Algebra Section 3.1 Problems

230

1. If the doughnuts in a bag consist of 6 chocolate doughnuts and 10 jelly- filled

doughnuts,

a) what is the ratio of jelly-filled to chocolate doughnuts reduced to lowest terms?

b) what is the ratio of chocolate doughnuts to the total number of doughnuts

in the bag? 2. Suppose a sports team plays 30 games, loses 10 of them, and ties none of

them.

a) What is the ratio of losses to games played?

b) What is the ratio of wins to losses?

3. What is the batting average of a baseball player who gets 15 hits in 60 times at bat?

4. If a baseball player has a batting average of 0.275 and has been at bat 2,000

times, how many hits has the player gotten?

5. Solve: x 7

10 80

6. Solve: 2 50

7 x

7. Solve: 1 x

6 72

8. Solve: 9 180

x 17

9. Solve: f t

g h for g

10. If 2 inches on a map represent 60 miles actual distance, how many miles do 5 inches represent?


231

11. If a car can travel 200 miles on 7 gallons of gasoline, how many miles can it

travel on 10.5 gallons of gasoline, assuming the same conditions? 12. If a 3 x 5 picture is enlarged so that the longer dimension is 20 inches, how long

will the shorter dimension be? 13. If 300 mL of alcohol must be used to make 500 mL of solution, how much

alcohol must be used to make 1250 mL of solution having the same strength? 14. If 25 g of salt will make 150 mL of solution, how much solution of the same

strength can be made using 40 g of salt? 15. If a temperature change of 100 Celsius degrees is the same as a temperature

change of 180 Fahrenheit degrees, what is the change in Celsius degrees if the temperature changes by 45 Fahrenheit degrees?

16. If the property tax on a $120,000 house is $900, what will be the tax on a

$200,000 house if the tax rate is the same for both houses? 17. If it takes 1.5 liters of orange juice to make punch for 10 persons, how much

orange juice will be required for 28 persons? 18. If a cement mixture calls for 60 pounds of cement for 3 gallons of water, how

much water must be used with 210 pounds of cement? 19. If 2 gallons of paint are needed to paint 75 feet of fence, how many gallons are

needed to paint 1350 feet of fence? 20. If a 140 pound person needs 2100 calories to maintain body weight, how many

calories does a 180 pound person need to maintain body weight, assuming both persons have the same activity level?

21. Suppose your living room measures 16 ft by 20 ft. If you wish to make a scale

drawing of the room, and the shorter side of the room is 12 in, how long will the longer side be?

22. Here is a medical problem involving proportions.

A patient is to receive 200 mg of cephalexin (Keflex), a medication. If the medication comes as a syrup containing 125 mg of the medication per 5 mL of syrup, how many mL of the syrup must be given?

Unit 3 Applications of Percent and Algebra Section 3.1 Answers to selected problems

232

1. a) 10 5

5 to 36 3

b)

6 6 33 to 8

10 6 16 8

2. a) 10 1

1 to 330 3

b)

30 10 20 2

2 to 110 10 1

3. batting average =

hits 15

0.250times at bat 60

4. Write 0.275 as a fraction. Then solve the proportion

275 x

1000 2000

by cross multiplying (275)(2000)=1,000x. Then divide to get

275 2000x 550

1000

5. 80x = (10)(7) 80x = 70

70

x 0.87580

6. 2x = (7)(50) 2x = 350

350

x 1752

7. 6x = (1)(72)

6x = 72 x = 12

8. 180x = (9)(17) x = 0.85

9. fh = gt

fh

gt

10. You are less likely to make a mistake on this type of problem if you write the units in the proportion.

Write 2 inches 5 inches

60 miles x miles

Then solve as follows: 2x= 300 x = 150 miles

Be sure to include units with your answer if units are given in the original problem.

11. 200 mi x mi

7 gal 10.5 gal

7x = (200)(10.5) 7x = 2100 x = 300 miles

12. shorter dimension 3 in x in

longer dimension 5 in 20 in

5x = 60 x = 12 inches


233

13. alcohol 300 mL x mL

solution 500 mL 1250 mL500 x = (300)(1250) x = 750 mL of alcohol

14. salt 25 g 40 g

solution 150 mL x mL

25x = (40)(150) x = 240 mL of solution

15. 100 degrees Celsius

180 degrees Celsius

x degrees Celsius

45 degrees Fahrenheit

180x = (100)(45) x = 25 Celsius degrees

16. tax $900 $x

value $120,000 $200,000

120,000x = (900)(200,000) x = $1500 tax

17. orange juice 1.5 L x L

persons 10 p 28 p

10x = (1.5)(28) x = 4.2 L

18. cement 60 lb 210 lb

water 3 gal x gal

60x = (3)(210) x = 10.5 gallons

19. paint 2 gal x gal

fence 75 ft 1350 ft

75x = (2)(1350) x = 36 gallons of paint

20. pounds 140 lb 180 lb

calories 2100 cal x cal

140x = (180)(2100) x = 2700 calories

21. shorter dimension 16 ft 12 in

longer dimension 20 ft x in

16x = (12)(20) x = 15 inches

22. 125 mg 200 mg

5 mL x mL

125 x = 1000 x = 8 mL

Unit 3 Applications of Percent and Algebra Section 3.2 Percent of difference

235

Objectives for this section Calculate the percent of difference between two numbers Calculate the change of a percent expressed in percentage points or percent of change Calculate the final amount when multiple percents of change are known using shortcut method

Relative Difference

The relative difference between two quantities is the difference between the quantities divided by one or the other of the two quantities. The quantity that appears as a denominator is called the reference quantity. The relative difference between two numbers may be expressed as a percent. This section examines several applications of percent of difference.

Calculating Percent of Change

Often, knowing just the amount by which a quantity changes isn’t very informative. A salary increase of $10,000 would be much more significant to a person who is presently earning $20,000 than it would be to a person who is presently earning $100,000. The relative change of a quantity compares the change in the quantity to the value of the quantity before the change, and is normally expressed as a percent. In other words, the relative change of a quantity uses the original or previous amount as the reference quantity.

Difference Between New Amount and Previous AmountRelative Change =

Previous Amount

The calculations of the relative changes for the two incomes mentioned above would be

10,000 10,000

0.5 50% and 0.1 10%20,000 100,000


236

Example 1. The temperature rises from 40 degrees to 60 degrees. What is the

percent of increase? Answer:

Change Difference Between New Amount and Previous Amount

Previous Amount Previous Amount

60 400.5

4050%

Example 2. The temperature falls from 60 degrees to 40 degrees. What is the percent of change? (It will represent a decrease.)

Answer:

60 40 20

0.333... 33.3% (approximately)60 60

Note that you divided by 60 in this case, because 60 is the temperature before the change. The correct way to state the answer is the temperature decreased by approximately 33.3%. Example 3. If a quantity becomes 4 times as large as it was previously, by what

percent has it increased? Answer: Suppose that the quantity was originally 1 and that it increased to 4,

which is four times as large. Then the percent of increase is:

4 1 3

1 1300%

Example 4. Suppose that a quantity decreases so that it is only one third of what it was originally. What is the percent of decrease?

Answer: Let the original amount be 3 and the final amount be 1, since 1 is one-

third of 3. Calculate as follows:

3 1 20.666...

3 366.6% (approximately)


237

The percent of change cannot be found if the original amount is zero.

Example 5. If the profit earned by a business increases from $0 per year to $50,000 per year, what is the percent of increase?

Answer:

New Amount Previous Amount 50,000 0

Previous Amount 0

which cannot be computed, because the calculation would involve division by zero.

Calculating Percent of Change When the Quantity That Changes is Already a Percent When the quantity that changes is itself a percent, confusion will arise unless the correct terminology is used. In such instances, we use the expression “percentage points” to describe the change in the quantity and the expression “percent of change” to describe the relative change in the quantity. As usual, the reference quantity is the percent before the change. The following examples illustrate the point:

Example 6. Suppose that an interest rate rises from 5% to 6%. Find the increase and also the percent of increase.

Answer: The increase is 1 percentage point (6% - 5%).

Change 1

0.20 20%Previous Amount 5

. This is the percent of increase.

The interest rate went up 20% when compared to the previous rate. In other words, a change of 1 percentage point is a 20% change when the change is compared to the original amount.


238

Example 7. The percent of smokers in a city decreases from 40% to 30% of the population. Find the percent of decrease.

Answer: The percent of smokers has decreased by 10 percentage points

(40% – 30%), but the relative change expressed as a percent is

10

0.2540

25% .

Therefore, the percent of smokers went down by 10 percentage points, or the percent of smokers decreased by 25 percent when compared to the previous figure.

Example 8. Ed's math test score increased from 50% to 80%. By what percent did his

grade increase? Answer: His grade has increased by 30 percentage points, which is a relative

increase of 30

50 or 60%.


239

1. A child's height increases from 50 inches to 54 inches. Find the percent of

increase. 2. If the 1994 population of China was 1,210,000,000 and the estimated

population in 2020 is 1,413,000,000, find the percent of increase. 3. The weight of person on a diet decreases from 240 pounds to 180 pounds. Find

the percent of decrease. 4. If your salary goes up from $4 per hour to $5 per hour, calculate the percent of

increase. 5. If your salary decreases from $5/hr to $4/hr, calculate the percent of

decrease. 6. If the percent of the public who use credit cards increases from 10% to 70%,

a) calculate the increase measured in percentage points.

b) calculate the percent of increase.

7. If the inflation rate decreases from 8% down to 2%,

a) calculate the decrease measured in percentage points.

b) calculate the percent of decrease.

8. If the percent of inflation in a country increases from 2% to 3%,

a) calculate the increase in percentage points.

b) calculate the relative increase as a percent.

9. If the percent of inflation in a country increases from 100% to 101%,

a) calculate the increase in percentage points.

b) calculate the relative increase as a percent.


240

10. If the price of gasoline drops from $2.09 per gallon to $1.79 per gallon,

a) calculate the decrease measured in dollars.

b) calculate the relative decrease measured in percent to the nearest tenth of a percent.

11. If the rate of inflation is presently 2%, and it increases by 1 percentage point,

calculate the new rate of inflation. 12. If the rate of inflation is presently 2%, and it increases by 50%, calculate the

new rate of inflation. 13. Can the percent of increase be more than 100%?

14. If a quantity triples, by what percent does it increase?

15. If interest rates rise from 4% to 6%, find the change

a) measured in percentage points. b) as a percent relative to the old interest rate.

16. If something becomes ten times as large as it was previously, by what percent has its size increased?


241

1.

54 50

0.08 8% increase50

2.

1,413,000,000 1,210,000,000

1,210,000,000

0.1678 = 16.78% increase

3.

240 180 60

0.25 25%240 240

decreases by 25%

4.

Change 5 4 1

Previous Amount 4 40.25 = 25% increase

5.

Change 5 4 1

Previous Amount 5 50.20 = 20% decrease

6. a) 70 - 10 = 60 percentage

point increase

b) 60

6 600%10

7. a) 8 - 2 = 6 percentage point decrease

b) 6

0.75 75%8

decrease

8. a) 1

b) 50%

9. a) 1

b) 1%

10. a) $0.30

b)

$2.09 $1.79 $0.30

$2.09 $2.09

14.4%

11. 2 + 1 = 3%

12. 2 + 0.5(2) = 3%

13. Yes. For instance, if a price increased from $5 to $15, the percent of increase would be

15 5 102 200%

5 5.

14. Use a numerical example: A

quantity triples by increasing from 100 to 300. Then the percent of increase is

300 100 2002 200%

100 100

15. a) 2 percentage points

b) 50%

17. 900%

Unit 3 Applications of Percent and Algebra Section 3.3 Applications of percent of difference

243

Objectives for this section Calculate the percent of discount Calculate the percent of markup based either on cost or on selling price Calculate either the original price or the final amount when the percent of change is known Calculate either the cost or the selling price when the percent of markup is known Calculate the original amount or the final amount when multiple percents of change are known using the shortcut method

Calculating Percent of Markdown (Discount)

A markdown or discount is what takes place when a business lowers the price of items to get rid of them, as in a sale. The percent of discount is normally a percent of what the item sold for originally. A discount is a decrease in the price of the item. The formula for calculating the percent of discount is

Difference Between Previous Price and Sale Price

Previous Price

and then change to a percent by multiplying by 100 or by moving the decimal two places to the right. Example 1. Calculate the percent of discount if an item is marked down from $120

to $90.

Answer: Difference Between Previous Price and Sale Price

Previous Price

120 90 300.25

120 12025%

The result would be called a 25% discount.

Example 2. An item selling for $800 is marked down to $400. Find the percent of markdown.

Answer: Percent of Markdown =

Difference Between Previous Price and Sale Price

Previous Price

800 400 4000.50

800 80050%


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Calculating Percent of Markup on Goods to be Sold

Markup (or margin) is the difference between what a business pays for an item and the selling price of the item. Since markup represents a change, the percent of markup may be found. Ordinarily the percent of change is found by dividing the change by the original amount. Finding the percent of markup differs from this procedure in that percent of markup may be calculated either by dividing the markup by the cost of the item or by dividing the markup by the selling price of the item. Often businesses use the selling price of goods to calculate the percent of markup (or margin), since the selling price is a quantity known to the person selling the goods. Hence, the percent of markup may be found by either of the following formulas depending upon the practice of the business: Markup = Selling Price - Cost

Selling Price Cost×100 based on cost

Cost

Percent Markup= OR

Selling Price Cost×100 based on selling price

Selling Price

The denominator when computing percent of markup may be either the cost or the selling price. This is a major difference from the previous problems on percent of change. Example 3. An item costing $50 is sold for $70. Find the percent markup based on

cost.

Answer: The cost is $50; therefore, we divide by 50.

70 50 20

0.450 50

40%

Example 4. Find the percent markup in the previous example if it is based on selling

price rather than cost.

Answer:

70 50 20

0.285714...70 70

28.6% (approximately)

Notice that the only difference between this example and the previous one is that the markup ($20) is divided by the selling price ($70), rather than the cost ($50).


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It can be seen from the previous two examples that the percent of markup based on cost is always larger than the percent of markup based on selling price. The formula relating the original price and the sale price:

Original Price - Discount = Sale Price

OR

Original Price - Percent of Discount (expressed as a decimal) x Original Price = Sale Price Note that these formulas are the same as the formulas for increase and decrease with a few words changed. The subtraction symbol in the formula tells you that the price has decreased rather than increased. Example 5. An item was originally marked $450. If it is later advertised at 20% off

the original price, what is the sale price? Answer: Original Price - Discount = Sale Price

Let x represent the Sale Price.

450 – (20%)(450) = x

450 – (0.2) (450) = x

450 – 90 = x

$360 = x

Example 6. After a 40% discount, an item sells for $360. What was the original price?

Answer: Original price – 40% of original price = Sale Price

Let x represent the original price.

x – 0.40x = 360

0.6x = 360

x = $600


246

Finding Cost or Selling Price when Percent of Markup (Margin) onGoods Sold is Known When the percent of markup is given, you must be careful to apply it to the appropriate quantity in a problem. If the percent of markup is based on cost, then it must be multiplied by the cost in a problem. If the percent of markup is based on selling price, then it must be multiplied by the selling price in a problem. The formula is: Cost + (Percent of Markup expressed as a decimal) x (either Cost or Selling Price)= Selling Price Example 7. An item costs a business $200. The markup is 30% based on cost. Find

the selling price. Answer: Let x represent the selling price.

Cost + Markup = Selling Price

Cost + 30% of cost = Selling Price

200 + (30%)(200) = x 200 + (0.30)(200) = x 200 + 60 = x $260 = x

Example 8. An item sells for $800. If the markup is 25% of the selling price, find the cost.

Answer: Let x represent the cost.


x + (25%)(800) = 800 x + (25)(0.01) (800) = 800 x + 200 = 800 x = 800 – 200 x = $600


247

Example 9. An item sells for $260. If the markup is 30% of the cost, find the cost. Answer: Let x represent the cost.


Cost +(30%)(Cost) = Selling Price

x + (30%)x = 260 x + 0.30x = 260 1.3x = 260

260

x1.3

x = $200

Example 10. The cost of an item is $500. Markup is 60% of the selling price. Find the selling price.

Answer: Let x be the selling price. Then the markup is 60% of x.


500 + (60%)x = x 500 + 0.6x = x 500 = x – 0.6x 500 = 0.4x

500

x0.4

$1250 = x

Remember: The dollar amount of the markup is always either the cost or the selling price multiplied by the percent of markup.

In Accounting, businesses always prepare a “Budget” for their next year of operations. In order to determine how well or how poorly the Budget was prepared, it is necessary to find the Percent of Difference between the Budget and the Actual results for the next year. We calculate this by finding the difference between the Actual and the Budget and then divide by the Budget. Example 11: If the Budget Revenue is $28,566 and the Actual Revenue is $32,583,

what is the percent of difference? Answer: Difference/Budget = ($32,583 - $28,566)/$28,566 = 14.06%.


248

Remember from Unit 1 that we learned a short cut for multiple increases/ decreases in a quantity. To increase or decrease a quantity by a percent, multiply it by 1 plus the percent of increase (expressed as a decimal) or 1 minus the percent of decrease (expressed as a decimal). The formula for change becomes:

Original Amount x (1 % of change expressed as a decimal) = New Amount

We are now going to apply this method to problems found in business:

Example 12. Suppose you earn $20,000, receive a 2% raise, followed by a 1% raise, followed by a 5% raise, followed by a 3% raise. How much will you earn after the last raise?

Answer: (old salary)(1 + 0.02)(1 + 0.01)(1 + 0.05)(1 + 0.03) =

(old salary)(1.02)(1.01)(1.05)(1.03) = new salary

(20,000) (1.02) (1.01) (1.05) (1.03) = $22,283.23

Example 13. Compare the result of getting a 20% discount followed by a 10% discount with the result of getting a single 30% discount. Assume that the original cost is $1000.

Answer: (1000)(1 – 0.20)(1 – 0.10) = 1000(0.8)(0.9) = $720 (separate)

(1000)(1 – 0.30) = 1000(0.7) = $700 (single)

The difference between getting both discounts simultaneously and getting them one after the other is $20.

Example 14. A $500 price is decreased by 20% and then further decreased by 30%.

What is the final result? Answer: The result is 500(0.8)(0.7) = $280.

Example 15. A $500 price is decreased by 20% and 30% simultaneously. What is the final result?

Answer: This amounts to a 50% reduction, and the result is 500(0.5) = $250.

Note the difference in the amounts from the two examples.


249

Example 16. Which is better, to receive a 10% raise, followed by a 10% salary cut, or to receive a 10% salary cut followed by a 10% raise?

Answer: It makes no difference. Suppose your salary is $1,000.

Raise followed by cut: (1000) (1.10) (0.90) = $990

Cut followed by raise: (1000) (0.90) (1.10) = $990

Note also that the raise and the cut do not cancel each other out even though they are the same percent. You always end up receiving less than your original salary if the raise and the cut are the same percent. In order for the raise and the cut to cancel out, the percent for the raise must be greater than the percent for the cut. This is demonstrated in the following examples. Example 17. 11% raise followed by a 10% cut

Answer: Assuming a starting salary of $1000:

(1000) (1.11) (0.90) = $999 You almost break even.

Example 18. 10% cut followed by 11% raise

Answer: Assuming a starting salary of $1000:

(1000) (0.90) (1.11) = $999 Same result.

To restate: When percent changes are applied to a quantity one after the other, the result is not the same as when the percent changes are all applied at the same time. However, the order in which percent changes are done makes no difference. An increase of a number by a certain percent followed by a decrease of the result by the same percent will not yield the original number.


250

Example 19. Mark up a price of $200 by 15%, and then reduce the result by 30%. Answer: New Price = (200) (1 + 0.15) (1 – 0.30) = (200)(1.15)(0.70) = $161. Example 20. Suppose that after a 20% discount followed by a 10% discount followed

by a 5% sales tax, an item costs $378. What was the original price of the item before any of the discounts or the sales tax?

Answer: Let x represent the original price.

(original price) (1 – 0.20)(1 – 0.10)(1+0.05) = 378

(x) (0.8)(0.9)(1.05) = 378

378

x(0.8)(0.9)(1.05)

x = $500

SUMMARY

1. When percent of markup is based on cost, the reference quantity is the cost. The formula relating cost, markup and selling price becomes:

Cost + (Percent of markup expressed as a decimal x Cost) = Selling Price

2. When percent of markup is based on selling price, the reference quantity is the selling price. The formula relating cost, markup and selling price becomes:

Cost + (Percent of markup expressed as a decimal x Selling Price) = Selling Price


251

1. Calculate the percent of discount if an item is marked down from $850 to $730. Round the answer to the nearest tenth of a percent.

2. An item selling for $4.50 is marked down to $1.80. Find the percent of

markdown. 3. If a refrigerator is originally priced at $1,600 and the amount of the discount is

$200, find the percent of discount for the refrigerator. 4. If a merchant sells an item which originally costs him $50 for $30, what is his

percent of loss on the sale? 5. If a pair of shoes goes on sale for $25 from the original price of $60, what is the

percent of discount for the pair of shoes? Round the answer to the nearest tenth of a percent.

6. The price of a car decreased by $2650 and was sold for $3500. What is the

percent of discount for the car? Round the answer to the nearest tenth of a percent.

7. A merchant sells an item originally costing $400 for $800. Calculate the

percent of markup based on cost. 8. In the previous question, calculate the percent of markup based on selling

price. 9. If an appliance costs a store $500 and sells for $800, calculate the percent of

markup based on cost. 10. In the previous question, calculate the percent of markup based on selling

price. 11. If the Advertising Budget for a company is $149, and the Actual Advertising

expense is $99.00, what is the percent of difference? 12. An item was originally marked $450. If it is later advertised at 20% off the

original price, what is the sale price? 13. After a 12% discount, an item sells for $308. What was the price before the

discount? 14. Some stock goes up in value by 35%. If it is now worth $621, what was it worth

before it went up?


252

15. An item costs $400. Markup is 80% of the cost. Find the selling price. 16. If a merchant pays $320 for an item and the markup is 20% based on selling

price, what will be the selling price of the item? 17. If an item costs a merchant $280 and the markup is 25% based on cost, what

will be the selling price of the item? 18. If a merchant sells an item for $338 and the markup is 30% based on cost, what

was the cost of the item? 19. If a merchant sells an item for $450 and the markup is 20% based on selling

price, what did the merchant pay for the item? 20. After a 10% discount and then a 5% sales tax, an item sells for $378. What was

the original price of the item? 21. If the original price of an item is $80, find the price after a 60% discount. 22. After a 40% discount, an item sells for $48. Find the price of the item before

the discount. 23. After a 10% salary increase, an employee earns $715 per week. Find the

weekly salary of the employee prior to the raise. 24. An item costs $30, and the markup is 250% of the cost. Find the selling price of

the item. 25. If the cost of a piece of merchandise is $300 and the markup is 40% of the

selling price, find the selling price. 26. If the selling price of an item is $350 and the markup is 30% of the selling price,

find the cost. 27. If the selling price of an item is $600 and the markup is 50% based on cost, find

the cost of the item. 28. If a piece of merchandise is discounted 50%, then discounted another 20%, and

then a 6% sales tax is added on, find the final price of the item if the original price is $200.

29. After a 25% discount, followed by a 10% discount, followed by an 8% sales tax,

the price of an item is $546.75. Find the original price of this item.


253

1. 850 – 730 = 120

120

14.1%850

2. 4.50 – 1.80 = 2.70

2.70

60%4.50

3. 200

12.5%1600

4. 50 – 30 = 20

20

40%50

5. 60 – 25 = 35

35

58.3%60

6. 2650 + 3500 = 6150

2650

43.1%6150

7. markup 400

1 100%cost 400

8. markup 400

0.5 50%selling price 800

9. 60% 10. 37.5% 11. -33.56%

12. 450 – (20%)(450) = x 450 – (0.2) (450) = x 450 - 90 = x $360 = x

13. x – 0.12x = 308 0.88x = 308

308

x $3500.88

14. x + 0.35x = 621 1.35x = 621

621

x $4601.35

15. 400 + (0.8) (400) = x 400 + 320 = x $720 = x

16. 320 + 0.2x = x 320 = x - 0.2x 320 = 0.8x

320

x0.8

$400 = x

17. 280 + (0.25)(280) = x 280 + 70 = x $350 = x

18. x + 0.3x = 338 1.3x = 338

338

x $2601.3

19. x + 0.2(450) = 450 x + 90 = 450 x = 450 - 90 x = $360

20. (x)(0.90)(1.05) = 378

378

x $400(0.90)(1.05)

21. $32 26. $245

22. $80 27. $400

23. $650 28. $84.80

24. $105 29. $750

25. $500

Unit 3 Applications of Percent and Algebra Section 3.4 Simple interest

255

Objectives for this section Solve for any one of the letters in the simple interest formula when the other three are known Solve for any one of the letters in the formula A = P + I when the other two are known Calculate average daily balance and use to calculate interest on credit card debt Calculate interest when rate is other than a yearly rate

Computing Simple Interest

When a person borrows money, the “rent” paid for the use of the money is called interest. How much interest is charged for the use of money depends upon how much money is borrowed, how long the money is kept, and what percent the institution making the loan charges.

The formula for computing interest is .

I is the amount of interest in dollars, P is the principal (amount borrowed) in dollars, R is the interest rate expressed as a decimal, and T is the time the money is kept. Usually the interest rate R is given as a percent and must be converted into a decimal before it is used in the formula. Example 1. You borrow $500 for 18 months at an interest rate of 12%. How much

interest do you pay on the loan? Answer: I = (500)(0.12)(1.5) = $90. Notice that 12% was changed to 0.12, and 18

months was changed to 1.5 years (divide by 12). Any one of the letters in the formula may be the unknown quantity as shown in the following examples: Example 2. If you pay $1200 interest on a loan of $4000 for 3 years, what is the

interest rate? Answer: 1200 = (4000).R.(3)

1200 = (12,000).R

1200

R12,000

0.10 = R 10% = R Change decimal to percent.


256

Example 3. If you pay $500 interest on a loan borrowed at 8% over a two-year

period, how much did you borrow? Answer: 500 = P.(0.08)(2)

500 = 0.16.P

500

P0.16

$3125 = P

Example 4. If you borrow $14,000 at 8% and pay $2800 interest, how long do you keep the money?

Answer: 2800 = (14,000)(0.08).T

2800 = 1120.T

2800

T1120 2.5 years = T

When you use the formula I = PRT, the assumption is that you are keeping the money for the entire time period, and that the interest rate is the rate for the time unit you are using for T. Normally an interest rate is the percent which must be paid for keeping money for one year. If the time period for the interest rate is not stated, it is assumed to be one year. If the interest rate is the rate per month, then T must be in months. If the interest rate is the rate per day, then T must be in days. Example 5. Suppose that the simple interest rate is 1.2% per month, and $500 is

borrowed for 6 months. How much interest will be charged when the loan is repaid?

Answer: I = (500)(0.012)(6)

I = $36

Note that in the example above, 6 months is not changed into 0.5 years, because the interest rate is “per month,” not “per year.”


257

The formula I = PRT correctly calculates interest based on the assumption that the person borrowing the money will keep it for the entire time period and not repay it in payments as is done with installment loans. The amount that must be paid

back to the lender is given by .

A is the amount repaid, and is sometimes called the future value of the loan.

P is the amount borrowed, and is sometimes called the present value of the loan.

I is the interest charged on the loan.

Example 6. If $800 is borrowed at 12% simple interest and paid back after 2 years, how much must be paid back to the lender?

Answer: First compute the amount of interest charged.

I = (800)(0.12)(2) I = $192

Then add the amount of interest to the principal. A = P + I A = 800 + 192 A = $992

Credit Cards and Average Daily Balance

Computing interest is a little more complicated for credit cards. Most credit card companies charge you no interest if you pay off your bill every month. (There are some exceptions to this. Credit cards which charge you interest even if you pay your bill in full at the end of the month are said to have no “grace period.”) The problem that arises when interest is charged is how to compute the principal. The method that is most often used is the “average daily balance” method. The amount that you owe a credit card company is not constant throughout the month. You may purchase with the card many times during the month, return merchandise, and make payments. To determine the amount on which you will be charged interest (assuming that you didn’t pay off your bill completely at the end of the previous month), the credit card company calculates how much you owe them at the end of each day of the billing period, adds all of these numbers together, and then divides by the number of days in the billing period. In other words, the company calculates the average amount that you owe them over the entire billing period, hence the name “average daily balance.”


258

Sum of outstanding balances for each day of billing periodAverage Daily Balance (ADB) =

Number of days in the billing period

Example 7. You have a balance of $200 on the first day of the billing period, and your balance remains $200 for the next 8 days. You then charge an $80 item, and your balance remains $280 for the next 5 days. You then return the $200 item, and your balance remains $80 for the next 4 days. You make a $50 payment, and your balance remains $30 for the remaining 14 days of the billing period.

Since you did not pay the amount you owed in full, how much money will the credit card company use for the principal when they charge you interest on your next statement?

Answer: First calculate the sum of all of the daily balances.

(200)(8) + (280)(5) + (80)(4) + (30)(14) = 3740

Then divide this by the number of days in the billing period.

3740

$120.6531

(rounded to the nearest cent)

So the Average Daily Balance is $120.65.

The credit card company would then use the simple interest formula, I = PRT, to calculate the interest you owe. Suppose the company is charging 16% per year.

In this case,

31(120.65)(0.16)

365I $1.64 . This is the amount of the interest or

finance charge. Example 8. Suppose that the average daily balance is $500 over a 30-day billing

period, and that the daily interest rate is 0.041096%. Find the finance charge for the 30-day billing period.

Answer: I = (500)(0.00041096)(30) = $6.16. The number of days (30) is used for

T, because the interest rate is a daily interest rate.


259

Below is an example of credit card “terms”. This is from July of 2014. Terms can change dramatically over a short period of time. Note the important terminology, “Annual Percentage Rate (APR),” “Penalty APR and When it Applies,” and various fees charged. Notice also that the APR varies based on the Prime Rate.

Read the fine print before obtaining or using a credit card.


260

1. Calculate the simple interest on a principal of $600 at 18% for a period of 2 years.

2. Calculate the simple interest on a principal of $2500 at 16.5% for a period of 42

months. 3. Assuming that interest is withdrawn when it is paid, calculate how long it will

take $20,000 to earn $1500 at 9% simple interest. 4. How much must be invested to earn $3000 simple interest over a 3-year period,

if the interest rate is 8%? 5. If $6000 earns $1600 simple interest over a 30-month period, calculate the

simple interest rate. 6. Suppose you borrowed $5000 for 9 months at 15% simple interest. How much

would you have to pay back at the end of 9 months, assuming that you make no payments prior to the end of 9 months?

7. If you borrowed $2000 for 6 months and paid back $2300 at the end of 6

months, what is the simple interest rate you paid? 8. A company collected $28,500 on a loan that was originally made four years ago

for $25,000. What was the simple interest rate of the loan? 9. In what amount of time does $25,000 earn $4125 simple interest at 4.5%?

10. What principal earns $15 simple interest in six months at 3%?

11. Suppose you owe the bank issuing a credit card $175 for the first 9 days of the billing cycle on which you are to be charged interest; you owe them $225 for the next 5 days of the cycle; you owe them $150 for the next 10 days of the cycle; and you owe them $270 for the last 6 days of the billing cycle. Compute your average daily balance and finance charge on this account for this billing cycle. Assume that the interest rate is 18%, and use 365 as the number of days in the year.


261

12. a) Suppose that you owe $800 on a credit card for the first 29 days of the billing cycle, and make a payment so that you owe $300 for the last 1 day of the billing cycle. How much interest would you pay if the interest rate is 24% using 365 as the number of days in the year?

b) How much interest would you have saved if you had made the payment

earlier so that you owed $800 for the first 19 days of the month and $300 for the last 11 days of the month?

13. Suppose that your average daily balance is $650, and that the daily interest

rate is 0.057534%. What would be the interest charges for a 29-day billing period?

14. How much interest would be charged on a loan of $100,000,000 for 3 days at an

interest rate of 9%, assuming 365 days in the year? 15. The Federal Reserve System of the United States charges banks interest for

every minute they borrow money. If the yearly interest rate is 2%, find the amount of interest paid by a bank which borrows $300,000,000 for 9 hours and 17 minutes. The number of minutes in a year is (365)(24)(60).

16. Find the interest which must be paid if $90,000 is borrowed for 6 months at 8%

simple interest and no payments are made prior to 6 months. 17. If the interest on a loan is $6000 over a period of 18 months and no payments

are made prior to 18 months, find the amount borrowed if the interest rate is 8%.

18. If $12,000 interest is paid on a loan of $300,000 borrowed for 3 months with no

payments prior to 3 months, find the simple interest rate.


262

1. I = ($600)(0.18)(2)

I = $216

2.

42I ($2500)(0.165)

12

I = $1443.75

3. 1500 = (20,000)(0.09)(T) 1500 = 1800T

1500

T1800

0.8333 years = T

0.8333 years x 12 months

year=

10 months

4. 3000 = (P)(0.08)(3) $12,500 = P

5.

301600 6000 R

12

10.67% = R

6.

9I 5000(0.15)

12

A = P + I A = $5562.50

7. 300 = (2000)(R)(0.5) 300 = 1000.R 0.30 = R 30% = R

8. 28,500 = 25,000 + I I = 3500 I = PRT 3500 = 25,000(R)(4) 3.5% = R

9. 4125 = 25,000(0.045)(T)

3.666667 yr = T Or 44 mo = T

10. 15 = (P)(0.03)

6

12

$1000 = P

11.

(175)(9) (225)(5) (150)(10) (270)(6)$194

30

ADB = $194

Finance charge:

I = (194)(0.18)

30

365

I = $2.87 (rounded to nearest cent)

12. a) $15.45 b) $15.45 - $12.16 = $3.29

13. I = (650)(0.00057534)(29) = $10.85

14. $73,972.60

15. I = (300,000,000)(0.02)

(9)(60) 17

(365)(24)(60) =$6,358.45

16. $3600

17. $50,000

18. 0.16 = 16%

Unit 3 Applications of Percent and Algebra Section 3.5 Installment Loans

263

Objectives for this section Calculate payment for add-on interest installment loans Calculate interest rate for add-on interest installment loans Convert interest rate R into annual percent rate APR for add-on installment loans Calculate payment for installment loans using annual percent rate

Add on Interest

Generally, when people borrow money, they obtain an installment loan. This type of loan is paid back in installments or payments every week, every month, every 3 months, every 6 months, or perhaps every year. The formula for simple interest, I = PRT, does not calculate the interest correctly for this type of loan. It correctly calculates the interest only in the case when no money is repaid until the end of the borrowing period. However, the formula may be used to a limited extent if the interest rate is corrected by yet another formula. The payment formula at the end of this section may be used to calculate the exact payment for an installment loan when the correct annual percentage rate is known. Some calculators have extensive capability to deal with such loans. Example 1. Suppose you borrow $800, and repay it over an 18-month period. The

interest rate quoted is an add-on rate of 7%. Add-on means that the interest is calculated using I = PRT as if you were not going to pay back any of the money until the end of the 18-month period.

What will be the amount of your monthly payment, and what is the true interest rate for this loan?

Answer: First compute the interest using I = PRT.

I = (800)(0.07)(1.5) = $84. Then add this amount to the principle. A = P + I = 800 + 84 = $884. You must repay $884 to the loan company.

Determine the number of equal payments, N.

There are 18 equal monthly payments.

Payment = A 884

N 18 $49.11 rounded to the nearest cent.


264

The true interest rate you paid in Example 1 is considerably more than 7%, since you did not keep all of the $800 you borrowed for the entire 18 months. As a matter of fact, you owed the loan company $800 for only one month. After the first month you have partially paid the loan back. By the time you make the last payment, you owe the company very little money.

To compute approximately the true interest rate you paid (called the Annual Percentage Rate or APR), use the formula

2 N RAPR

N 1

where N is the number of (equal) payments you make, and R is the interest rate used

in the formula I = PRT. In Example 1, the Annual Percentage Rate (APR) is:

(2)(18)(0.07)APR

(18 1)

13.26%

Notice that the APR is nearly double the add-on interest rate. By federal law, the APR must be stated on practically every loan. Notice that APR depends only upon the add-on interest rate, R, and upon the number of payments, N. Using a financial calculator rather than the formula to find the APR in this example yields 12.87%. If you buy an item and make a down payment, the down payment is subtracted from the price you pay before the finance charges are added. The down payment is the amount you pay when you buy the item. An example is your old car that you trade in when you buy a new car.

Example 2. You purchase a car which costs $18,000 and pay $3000 down. How much money do you have to borrow?

Answer: $15,000, since $18,000 - $3000 = $15,000.

The finance charges, or interest, will be calculated using $15,000, since you borrowed only $15,000, not $18,000.


265

Example 3. Here is an example using most of the interest formulas: You purchase a

car which costs $18,000. For the down payment, you trade in a car worth $4000. You make 48 monthly payments of $350 each.

a) Calculate the amount of money borrowed. b) Calculate the total amount you pay back to the lender. c) Calculate the amount of interest you pay to the lender. d) Calculate the add-on interest rate. e) Calculate the true interest rate (APR) approximately. f) Calculate the total amount you paid for the car.

Answers: a) P = Cost - Down payment = 18,000 – 4000 = $14,000

b) A = Number of payments x Amount of payment = (48)(350) = $16,800

c) I = A - P = 16,800 – 14,000 = $2800

d) I = PRT 2800 = (14,000)(R)(4) 2800 = 56,000.R

2800

R56,000

0.05 = R 5% = R

e) (2)(48)(0.05)

APR(48 1)

9.8%

Remember that this is only an approximation to the correct APR. A financial calculator yields a result of 9.24%.

f) Total paid = amount paid to lender + down payment

Total paid = 16,800 + 4000 = $20,800

If you pay off an installment loan early, you may find that you save very little interest. Most of the interest is paid with your earlier payments, and very little is paid with your later payments. There are some financial formulas for calculating how much interest you save by paying off a loan early.


266

Calculating the payment directly when the APR is known

Using the simple interest rate, R, and the simple interest formula to calculate the payment on an installment loan is a rather cumbersome method, since the APR is the interest rate which is usually known. Here is a way to use the APR directly to calculate your payment on an installment loan. This method gives exact values.

-n×T

P amount borrowedAPRP

APR annual percentage ratenPayment= where

n number of payments per yearAPR1 1

T loan term in yearsn

Note that when using the formula daily payments n = 365 weekly payments n = 52 monthly payments n = 12 quarterly payments n = 4 semi-annual payments n = 2

T is always the number of years, whether a whole number or a fraction.

The quantity, (n.T) (n times T), is the total number of payments over the life of the loan. Example 4. Find the payment amount if $150,000 is borrowed for 15 years at an APR

of 5.5%, and payments are made monthly.

Answer: Payment =

12 15

0.055150,000

12

0.0551 1

12

$1,225.63 each month


267

Example 5. Find the payment amount if the APR is 15%, and $2000 is borrowed for 20 weeks to be repaid weekly.

Answer: Payment =

2052

52

0.152000

52

0.151 1

52

$103.06 each week

Example 6. Find the payment amount if the APR is 24%, and $3000 is borrowed for 18 months and repaid monthly.

Answer: Payment =

1812

12

0.243000

12

0.241 1

12

$200.11 each month

Example 7. How much must be paid back each quarter if $80,000 is borrowed at an APR of 6% for a period of 5 years?

Answer: Payment =

$4659.66 each quarter4 5

0.0680,000

4

0.061 1

4

Example 8. Suppose that $75,000 is borrowed for 4 years at an APR of 8%, and

payments are made semi-annually. Find the payment.

Answer: Payment =

2 4

0.0875,000

2

0.081 1

2

$11,139.59 semi-annually =

$11,139.59 semi-annually


268

Use the APR formula for converting between the add-on interest rate and the Annual Percentage Rate (the actual interest rate paid). Remember that the formula is only an approximation; therefore, your answers will also be approximations to the actual value. 1. A car company is offering car loans at an add-on interest rate of 12 percent,

and you buy a car for $6000 with no money down and finance it for 24 months.

a) Find the amount of interest you will pay.

b) Calculate your monthly payment.

2. Find the amount of interest and monthly payment for each of the following loans. All interest rates are add-on rates.

a) $ 2500 loan at 12% for 2 years

b) $ 5000 loan at 6% for 15 months

c) $4300 loan at 9.5% for 3 years

d) $1800 loan at 21% for 27 months

3. Find the approximate APR (to the nearest tenth of a percent) for each of the following situations assuming monthly payments:

a) Purchase a stereo for $2400 at 18% add-on interest for 3 years.

b) Purchase a microwave for $600 at 15% add-on interest for 1 year.

c) Purchase a car for $18,500 at 6% add-on rate for 5 years.

d) Purchase a hot tub for $4500 at 16% add-on rate for 2 years.

4. Libby purchased a stereo for $1800. She bought the stereo on the installment plan, paying $300 down, and agreeing to pay the balance in 12 monthly payments. The finance charge was 16% add-on interest on the balance. Find the approximate annual percent rate to the nearest tenth of a percent.

5. Using problem #4, compute the amount of interest paid and the amount of the

monthly payment.


269

6. A VCR is advertised for $500. It can be purchased on the installment plan for

$50 down and an agreement to pay 15% add-on interest on the balance over 12 months.

a) Find the approximate APR to the nearest tenth of a percent.

b) Find the finance charge.

c) Find the total cost of the VCR?

7. Suppose you buy an appliance which costs $650, pay $50 down, and then make payments of $40 per month for 18 months.

a) How much interest do you pay?

b) Calculate the add-on interest rate.

c) Find the approximate APR.

d) What is the total amount you pay for the appliance?

8. Suppose you buy a car which costs $18,500, pay $3,500 down, and then make 48 monthly payments of $380 each.

a) How much interest do you pay?

b) Find the approximate APR.

9. Suppose the APR is 16.2%, and you borrow $10,000 for 4 years, and make monthly payments.

a) What is the exact amount of your payment to the nearest cent?

b) What is the total amount you pay back using the payment from a)?

c) How much interest do you pay?

10. To help pay her tuition bill, Monica borrowed $1100 at 9.5% simple interest for a period of 18 months. How much interest will she pay?


270

11. A newspaper advertisement offers a $22,500 car for nothing down and 60

monthly payments of $630.

a) What is the total amount paid for both the car and the financing?

b) What is the simple interest rate assuming add-on interest? (Round to the nearest tenth of a percent.)

12. A furniture dealer would like to sell you a sofa for $1200, and to finance it for

12% add-on interest for 36 monthly payments of $45.33.

a) Find the interest paid.

b) Find the approximate APR using the formula from our text.

13. A car company is offering car loans at a simple interest rate of 8.5%. Find the amount of interest charged to a customer who finances a car loan of $7200 for 30 months.

14. A car dealer will sell you an $18,500 car for nothing down and 60 easy monthly

payments of $450.

a) What is the total amount paid for both the car and the financing?

b) What is the simple interest rate (to the nearest tenth of a percent)?

15. You would like to buy a computer for $1800 and to finance it for 10% add-on interest for 42 monthly payments of $57.86. Find the approximate APR using the formula in the text.

16. A company is expanding its line to include more products. To do so, it borrows

$320,000 at 13.5% simple interest for a period of 18 months. How much interest must the company pay?

17. Susan is investigating two different ways to pay for a new car on which she

owes $3000. Which of these two plans for financing the car gives the best APR?

a) Her bank offered to let her borrow the money for 6.5% add-on interest for 2.5 years with equal monthly payments.

b) Another bank offered her 6% add-on interest for 3 years with equal monthly

payments.


271

18. Marty obtained $190 from the bank, and signed a 3-month note to pay back $200. Compute the simple interest rate he was charged (to the nearest tenth of a percent).

19. Calculate the exact monthly payment (to the nearest cent) if $15,000 is

borrowed at an APR of 8% for a term of 5 years. 20. Calculate the exact monthly payment (to the nearest cent) if $9000 is

borrowed at an APR of 10% for 42 months. 21. Calculate the exact payment (to the nearest cent) if $300,000 is borrowed at

an APR of 7% for 10 years, and payments are made every three months. 22. Calculate the exact payment (to the nearest cent) if $2000 is borrowed at an

APR of 30% for one year, and payments are made weekly. 23. Calculate the exact daily payment (to the nearest cent) if $500 is borrowed at

an APR of 600% for 30 days.


272

1. a) I = $6000(0.12)(2)

I = $1440

b) Monthly payment =

6000 1440$310

24

2. a) I = $600 payment = $129.17

b) I = (5000)(0.06)

15

12

I = $375 Monthly payment =

5000 375$358.33

15

c) I = $1225.50

Monthly payment = $153.49

d) I = $850.50 Monthly payment = $98.17

3. a) (2)(36)(0.18)

APR 35%37

b) APR 27.7%

c) APR 11.8%

d) APR 30.7%

4. (2)(12)(0.16)

APR 29.5%13

5. I = (1500)(0.16)(1) = $240 payment = $145

6. a) APR 27.7%

b) (450)(0.15)(1) = $67.50

c) $500 + 67.50 = $567.50

7. a) $120

b) 13.3% (rounded to one decimal place)

c) 25.3%

(rounded to one decimal place)

d) $770

8. a) $3240

b) 10.6% (rounded to one decimal place)

9. a) $284.43

(rounded to nearest cent)

b) $13,652.64

c) $3652.64

10. $156.75

11. a) $37,800 b) 13.6%

12. a) $431.88 b) 23.4%

13. $1530


273

14. a) $27,000 b) 9.2%

15. 19.5%

16. $64,800

17. a) 12.6% b) 11.7%

b) has the lowest APR

18. 21.1%

19. $304.15

20. $254.85

21. $10,491.63

22. $44.63

23. $21.25

Unit 3 Applications of Percent and Algebra Section 3.6 Compound interest

275

Objectives for this section Identify and calculate both future value and present value for compound interest problems Identify and calculate both future value and present value for inflation problems

Calculating Compound Interest

When interest is allowed to remain in an account, and future interest is paid on both the original principal and the accumulated interest, the process is called compounding. Example 1. Suppose that an investment of $1000 pays 12% interest at the end of

each year. Calculate the amount in the account at the end of the second year if the interest was paid as simple interest rather than as compound interest.

Answer: Use I = PRT = (1000)(0.12)(2) = $240.

Then use A = P + I = 1000 + 240 = $1240 to find the amount.

Example 2. Find how much the investment will be worth at the end of two years assuming that the interest paid in the first year is not withdrawn. This is called compounding.

Answer: To find the amount in the account at the end of the first year, multiply

by 1.12 (remember to increase 12%--multiply by (1 + 0.12). This gives the answer (1000)(1.12) = $1120. Since the interest is not withdrawn, the investment starts the following year with $1120 rather than $1000. So the amount at the end of the second year will be (1120)(1.12) = $1254.40. Note that this amount is greater than the simple interest amount by $14.40.

Example 3. Find how much the investment will be worth at the end of two years if

the interest is paid semi-annually (twice a year) assuming that nothing is withdrawn.

Answer: Since the interest is being paid twice a year, only 6% will be paid each

time (half of 12%). Thus, 1000 must be multiplied by 1.06 every six months. This must be done 4 times over a two-year period or (1000)(1.06)4, which is $1262.48 when rounded to the nearest cent.


276

Example 4. To extend the previous example, calculate how much the $1000 investment will be worth at the end of two years if the interest is compounded quarterly (paid four times per year and not withdrawn).

Answer: If the 12% interest is paid four times per year, only 3% (3 months’

interest) would be paid each time, 12%

3%4

. Calculating the amount in

the account would involve multiplying 1000 by 1.03 eight times over a two-year period, because there are eight quarters in two years. A simple way to do this is to calculate A = (1000)(1.03)8 = $1266.77

Notice that the amount here is greater than in the previous example by $4.29.

Example 5. Calculate the amount in the investment if the interest is compounded

monthly (paid twelve times per year).

Answer: In this case only 1% would be paid each month, 12%

1%12

. Adding this

percent to 1 yields 1.01. Over a two-year period, the investment would be multiplied by this number twenty-four times.

Calculating 24A (1000)(1.01) $1269.73 , an increase of $2.96 over the

previous example. Example 6. Calculate the amount in the investment if the interest is compounded

daily (paid 365 times per year). Answer: The interest rate must be divided by 365, and the interest must be paid

(2)(365) times over a two-year period. Thus the calculation is

365 20.12

A 1000 1365

$1271.20

This is only a slight increase of $1.47 over the previous example. So banks seldom compound interest more often than daily.


277

From the preceding examples we see that the formula for compounding is

n×T

RA P 1

n where

A is the amount in the account at the end of the time period. P is the principal or amount originally invested. R is the interest rate expressed as a decimal. n is the number of times per year the interest is paid. T is the time in years.

Sometimes A is called the future value of the investment, and P is called the present value of the investment. In any event, the amount in the investment at a later time is always called A, and the amount in the investment at an earlier time is always called P. Example 7. The following chart, taken from previous examples, illustrates the value

of compounding for

P = $1000, R = 12%, and T = 2 years.

Compounded Value of n Amount (A)

none (simple interest) does not apply $1240

annually 1 $1254.40

semi-annually 2 $1262.48

quarterly 4 $1266.77

monthly 12 $1269.73

daily 365 $1271.20

Compound interest is calculated by using the formula I = A – P, since we have given no formula for directly computing compound interest as we did for simple interest.


278

Calculating Present Value

Sometimes we must solve for P (present value) in the formula.

Example 8. What salary in 1960 was equivalent to a salary of $50,000 in 1990? Assume a 6% average yearly inflation rate.

Answer: T = 30, because this is a thirty-year period; A = 50,000, because this is

the amount at a later time; n = 1 for inflation; and P is the 1960 salary we are trying to calculate. Substituting the numbers into the formula gives:

50,000 = (P)(1.06)30

Solving for P by dividing both sides by (1.06)30 gives

30

50,000

1.06$8705.51 P .

Thus, a 1960 salary of $8,705.51 is equivalent to a 1990 salary of $50,000, provided the average inflation during the period from 1960 to 1990 was 6%.

Example 9. How much must be invested at 6% compounded quarterly in order to have $30,000 in an account 15 years from now, assuming that there are no withdrawals?

Answer:

4 150.06

30,000 P 14

Solving for P yields P = $12,278.88 rounded to the nearest cent.


279

Inflation

The compound interest formula can also be used to investigate the effects of inflation provided the rate of inflation is constant. Inflation is the tendency for the prices of goods and services to rise or inflate. When solving problems involving inflation, assume n = 1 in the compound interest formula. The inflation rate is the percent by which the average price of goods and services rises from one year to the next. For instance, if the price rises from $1000 one year to $1050 the next year, then the

inflation rate is the fraction 50

1000 expressed as a percent or 5%.

Example 10. If we assume an inflation rate of 4% over the next 10 years, what will a

salary of $30,000 today need to be 10 years from now to keep up with inflation?

Answer: 10A 30,000(1.04) $44,407.33

Therefore, in ten years the person who earns $30,000 today will need to earn $44,407.33 just to retain the same buying power as today, assuming that there is 4% inflation for the next 10 years.

A fundamental principle of Present Value is that money to be received in the future is not worth as much as money received today if there is inflation. Example 11. What is the present value of $4000 to be received 5 years from now if

inflation is assumed to be 3% per year for the next 5 years? Answer: Once again, solve for P in the compound interest formula, assuming that

inflation occurs once a year.

5

4000

1.03$3450.44 P (to the nearest penny)

Note: The compound interest formula is not used to compute the value of loans or of investments where the interest is withdrawn when it is paid.

You must determine whether future value or present value is to be calculated when solving a compound interest problem.


280

1. If you invest $2000 at 6% interest compounded semi-annually, how much will be in the account after 5 years? Assume that you don’t withdraw anything from the investment or add anything to the investment during this period.

2. In the previous problem, how much interest will the investment earn?

3. If a business invests $3,000,000 at 10% interest compounded daily for a period of 1 year, how much interest is earned?

4. How much is a 1950 salary of $10,000 worth in 1997 assuming a yearly inflation

rate of 6%? 5. What is the present value of $50,000 which you are to receive 20 years from

now, assuming that the inflation rate for the next 20 years will be about 4% per year?

6. How much must you invest today in order to have $200,000 thirty years from

now if you can earn 6% interest compounded monthly? 7. The Merritt family wishes to have $60,000 available for their newborn child’s

college education. How much should they invest today in a savings account that pays 9% interest compounded semi-annually to accumulate the needed amount by the end of 18 years?

8. Larry wants to have $9000 in his account in four years so that he can buy a car.

How much must he invest now at 12% compounded quarterly to accumulate the $9000?

9. Annette wishes to have $20,000 in her account in four years to have a down

payment for a piece of property. How much must she invest now at 10% compounded semi-annually to accumulate the $20,000?

10. A child’s grandparents wish to purchase a bond fund that matures in 18 years to

be used for the grandchild’s college education. The bond fund pays 4% interest compounded semi-annually. How much should they purchase so that the bond fund will be worth $85,000 at maturity?

11. You have just inherited a diamond and ruby ring appraised at $5000. If the ring

has appreciated in value at a rate of 9% compounded annually, what was the value of the ring 12 years ago when it was purchased?

12. How much must be invested today in order to have $25,000 ten years from now

if the money can earn 9% interest compounded monthly?


281

13. If $3000 is invested today at 7% interest compounded quarterly, how much will

be in the account 6 years from now? 14. If a person receives a 3% raise every six months, how much will the person earn

per year fifteen years from now if the present salary is $22,000 per year. 15. If the population of a country grows 2% per year, what will be the population of

this country thirty years from now if the present population is 100,000,000 persons? Assume compounding takes place just once per year.

16. Calculate how much more interest daily compounding earns over a one year

period than annual compounding if an account contains $50,000 and the interest rate is 10%.

17. Which is the better deal, 6% compounded daily or 6.15% compounded annually? 18. An inflation table available on the Internet shows that a nickel would inflate to

one dollar in the period from the year 1900 to the year 2000. What has been the average yearly inflation rate to the nearest hundredth of a percent over this 100-year period? This problem requires considerable use of your calculator and a fair amount of trial and error.

Writing Exercises

19. If you are investing money in a savings account paying a certain interest rate, which account should you choose, one compounding 4 times per year or one compounding 10 times per year? Explain your answer.

20. The information in this problem came from a business magazine. The average

cost of a 4-year degree for students beginning college in the year 2005 was $252,000 at an expensive private college. When John’s son David was born in 1987, John made an investment large enough for his son to attend the private college in 2005. If the investment earns 6% compounded annually, how much did he invest?

21. Explain the difference between simple interest and compound interest.


282

1.

(2 5)0.06

A 2000 12

= $2687.83

2. A - P = $ 2687.83 - 2000 = $687.83

3.

(365 1)0.10

A 3,000,000 1365

= $3,315,467.34

I = A - P = $315,467.34

4. 1997 - 1950 = 47 years

47A 10,000(1 0.06)

= $154,659.17

5. 4750,000 P (1 0.04)

20

50,000P

1.04

$22,819.35 = P

6.

12 300.06

200,000 P 112

12 30

200,000P

0.061

12

$33,208.39 = P

7.

2 18

60,000$12,301.69

0.091

2

8.

4 4

9000$5608.50

0.121

4

9.

2 4

20,000$13,536.79

0.101

2

10.

2 18

85,000$41,668.97

0.041

2

11.

1 12

5000$1777.67

0.091

1

12.

12 10

25,000$10,198.43

0.091

12

13.

4 60.07

3000 14

$4549.33

14. 3022,000(1.03) $53,399.77

15. 30100,000,000 (1.02)

181,136,158

16. 50,000(1.10) $55,000

3650.10

50,000 1365

$55,257.79

$257.79 more

17.

3650.06

1 1.06183365

0.06151 1.0615

1

6% compounded daily is a little better.

Unit 3 Applications of Percent and Algebra Section 3.7 Unit review

283


I = PRT I = interest P = principal R = interest rate T = time

A = P + I A = amount including interest

Payment = A

N N = total number of payments

Compound interest

1

n T

RA P

n

n = number of times per year money is compounded

yearly n = 1 semi-annually n = 2 quarterly n = 4 monthly n = 12 weekly n = 52 daily n = 365


2 N RAPR

N 1

(approximately) N = total number of payments

APR = annual percentage rate

Amount of payment for an installment loan where the APR is given

-n T

P amount borrowedAPRP




Unit 3 Applications of Percent and Algebra Section 3.7 Unit review problems

284

1. Find the size of the gasoline tank of a pickup which can travel 360 miles on a tank of gasoline if the pickup gets 25 miles per gallon.

2. If we use 80 grams of salt to make 500 milliliters of a solution, find how many

grams of salt must we use to make 1250 milliliters of the same strength solution.

3. A room contains 30 Germans, 20 Japanese, and 40 Nigerians.

a) Find the simplified ratio of Nigerians to Germans in the room.

b) Find the simplified ratio of Germans to the total number in the room.

4. If 80 pounds of food are required to feed 50 persons, find how many pounds will be required to feed 120 persons.

5. Henry's hourly wage dropped from $16 per hour to $12 per hour. Find the

percent by which his wage decreased.

6. From year one to year two, the equipment of a small company decreased in value from $15,000 to $0. Calculate the percent of difference in the value of the equipment.

7. Your salary is $250 per week. What will you earn after an 8% raise?

8. The price of a car increased by $1080. This was a 9% increase.

a) Find the previous price of the car.

b) Find the price of the car after the increase.

9. A merchant buys an item for $0.40 and sells it for $1.50.

a) Find the dollar amount of the markup.

b) Find the percent of markup based on cost.

c) Find the percent of markup based on selling price.

10. After a 9% raise, Renee earns $43,055.

a) Set up the equation to find her salary before the raise.

b) Solve the equation to find her salary before the raise.


285

11. An item costs a merchant $2340, and the markup is 35% of the selling price.

a) Set up the equation to find the selling price.

b) Solve the equation, and find the selling price.

12. An item is discounted 25%, then further discounted 10%, and finally an 8% sales tax is added on.

a) If the original price is $1200, find the final price to the nearest cent.

b) If the final price is $900, find the original price to the nearest cent.

13. The interest on a 6.5% add-on loan for six years is $1014.

a) Find the amount borrowed.

b) Find the amount which must be paid back.

14. If you get a $2000 loan with an 8.5% add-on interest rate for 42 months,

a) calculate the amount of interest you must pay.

b) calculate the amount of your monthly payment.

15. A car dealer will sell you a $19,600 car for $3500 down and 48 monthly payments of $410.75.

a) Calculate how much interest you pay.

b) Find the APR.

c) Find the total amount you will pay for the car.

16. If your credit card uses the average daily balance method to compute finance charges, and your balances are $900 for 7 days, $500 for 13 days, and $300 for 11 days,

a) Find your average daily balance for the period.

b) Compute the finance charges for the period, assuming 365 days in the year and an interest rate of 16.5%.


286

17. Assume that inflation has averaged about 5% per year. A salary in 1980 of

$10,000 would be equivalent to what salary in 1997? Assume that inflation occurs once a year.

18. If we wish to have $20,000 fifteen years from now and can earn 9% interest

compounded monthly a) Find how much we should invest now.

b) Find the interest earned over the 15-year period.

19. Which will yield a greater amount, money invested at 8.25% compounded once a year or money invested at 8% compounded daily, assuming that the amount invested and the time periods are the same for both investments?

20. Can the percent of markup based on selling price be more than 100%?

21. Can the percent of markup based on cost be more than 100%?

22. If inflation is 10%, use your calculator to determine how long it will take prices to triple, assuming inflation occurs once a year.

23. If the percent of persons attending two-year colleges increased from 25% of the

population to 35% of the population, a) find the increase in percentage points.

b) find the percent of increase.

24. If $4000 is borrowed at an APR of 18% for a period of 8 months and payments are made monthly, find the amount of the payment.

25. If $60,000 is borrowed at an APR of 7% for a period of 5 years and payments are

made quarterly (every 3 months), find the amount of the payment.


287

1. 14.4 gallons

2. 200 grams

3. a) 4

3 b)

1

3

4. 192 lbs

5.

16 12 4

25%16 16

6. ($0 - $15,000)/$15,000 = 100% decrease

7. 250(1.08) = $270

8. a) 0.09x =1080 x = $12,000

b) 12,000 + 1080 = $13,080

9. a) 0.40 + M = 1.50 M = $1.10

b) 1.10

275%0.40

c) 1.10

73.3%1.50

10. a) x + 0.09x = 43,055

b) 1.09x = 43,055 x = $39,500

11. a) 2340 + 0.35S = S

b) 2340 = S - 0.35S 2340 = 0.65S $3600 = S

12. a) 1200(0.75)(0.90)(1.08)=

$874.80 b) 900 = x(0.75)(0.90)(1.08)

900x

(0.75)(0.90)(1.08)

$1234.57 = x

13. a) I = PRT 1014 = (P)(0.065)(6) $2600 = P

b) $2600 + $1014 = $3614

14. a) I = (2000)(0.085)(3.5) I = $595

b)

(2000 595)

$61.7942

15. a) A = (48)(410.75) = $19,716 P = $19,600 – $3500 = $16,100 I = $19,716 – $16,100 = $3,616

b) I = PRT 3616 = (16,100)(R)(4) 5.6% = R

(2)(48)(0.056)APR 11%

49

c) Total = A + down payment = $19,716 + $3500 = $23,216


288

16. a)

(900)(7) (500)(13) (300)(11)

(7 13 11)

$519.35

b)

31(519.35)(0.165) $7.28

365

17. 1710,000(1 0.05) $22,920.18

18.

a)

(15 12)0.09

20,000 P 112

$5,210.99 = P

b) I = 20,000 - $5,210.99 = $14,789.01

19. 8.25% compounded once a year, multiply by 1.0825 each year.

8% compounded daily, multiply by

3650.08

1 1.0833365

each year.

Hence, 8% compounded daily is better, because 1,0833 is a larger number than 1.0825.

20. No

21. Yes

22. Take 1.1 to higher and higher

powers until you get three. Answer is between 11 and 12 years.

23. a) 10 percentage points

b) 10

0.40 40%25

24. P = $4000 APR = 18% n = 12

T = 8

12

812

12

0.184000

12Payment=

0.181 1

12

= $534.34

25. P = $60,000 APR = 7% n = 4 T = 5

4 5

0.0760,000

4Payment=

0.071 1

4

= $3581.47

289

UNIT 4

GEOMETRY AND UNITS OF MEASUREMENT

Unit 4 Geometry and Units of Measurement Section 4.1 Introduction, definitions, angles

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Objectives for this section Identify types of angles Identify different types of polygons Find unknown angles in triangles

Geometry

Geometry is the study of shapes and their properties. In this unit we will study both the shape and size of objects. Before proceeding, be sure you know what is being discussed when the following terminology is used.

Geometric Terminology

Example 1. point: A point marks a position and has no length or width.

It is represented by a period and labeled with a letter as shown.

Example 2. line: The term line normally means a geometric object which is straight and infinitely long.

A line is represented by two points on the line each labeled by letters as

shown and written ABsuur

. Example 3. line segment: A line segment is a section of a line that has a finite length

such as 20 cm or 2 feet.

A line segment is represented by the two endpoints of the section as shown

and written AB . Example 4. ray: The portion of a line which extends indefinitely in one direction but not

in the other direction.

A ray is represented by one endpoint and another point on the line as shown

and written

uuurAB .


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Example 5. angle: An angle is a figure formed by two rays meeting at a common endpoint called the vertex of the angle.

An angle is represented (in order) by one point on one ray, the vertex, and

the third point on the other ray. The representation will be written ABC

or simply B for the angle shown. The middle letter is always the vertex of the angle if three letters are used.

Example 6. degree: A degree is one of the units used to measure the size of an angle. Three-hundred-sixty (360) degrees (written 3600) make up a complete revolution. This is the unit we will use in this course to measure angles.

Example 7. straight angle: A straight angle is a line that measures 1800.

Example 8. right angle: A right angle is an angle which measures 90o. If two right angles are put together, then two of their sides form a line.

Example 9. acute angle: An acute angle is an angle which measures less than 900.

Example 10. obtuse angle: An obtuse angle is an angle which measures more than 900 but less than 1800.

Example 11. vertex: The vertex is the point on a polygon or an angle where two sides

meet.


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Example 12. polygon: A polygon is a closed figure made of line segments connected at

their endpoints. Each corner is called a vertex (plural is vertices).

Example 13. triangle: A triangle is a three-sided polygon.

Example 14. isosceles triangle: An isosceles triangle is a triangle with two equal sides.

Example 15. equilateral triangle: An equilateral triangle is a triangle with three equal sides.

Example 16. quadrilateral: A quadrilateral is a four-sided polygon.

Example 17. trapezoid: A trapezoid is a quadrilateral with two of its opposite sides parallel.


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Example 18. parallelogram: A parallelogram is a quadrilateral with its opposite sides parallel. Also, the parallelogram has opposite sides that are equal. The angles of a parallelogram do not have to be right angles like those of a rectangle.

Example 19. rhombus: A rhombus is a quadrilateral with four equal sides. The angles of a rhombus do not have to be right angles.

Example 20. rectangle: A rectangle is a parallelogram with four right angles. The opposite sides have the same length.

Example 21. square: A square is a rectangle with four equal sides.


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Example 22. pentagon: A pentagon is a five-sided polygon. As an example, the US military headquarters, “The Pentagon,” has the shape of a pentagon.

Example 23. hexagon: A hexagon is a six-sided polygon. For example, a beehive cell or the head of bolt has the shape of a hexagon.

Example 24. octagon: An octagon is an eight-sided polygon. For example, a stop sign has the shape of an octagon.

Example 25. circle: A circle is the set of points which are the same distance from some fixed point called the center of the circle.

Example 26. radius: The radius is the distance from the center of a circle to the circle. Also it can be described as a line segment from the center of the circle to the circle.

Example 27. diameter: The diameter of a circle is the distance from one side of a circle

to the other side through the center of the circle. Also it can be described as a line segment from one side of a circle to the other side through the center of the circle. The length of the diameter is twice that of the radius.


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Example 28. prism: A prism is a three dimensional figure whose top and bottom are identical and parallel to each other. In addition, the edges of the faces of a prism are parallel. An easy example of a prism is a standard room in a house; the floor and the ceiling are parallel to each other and have the same shape.

Note: Lines which would not be visible on a three dimensional figure are called hidden lines and are dotted rather than solid.

Example 29. cylinder: A cylinder is a three dimensional figure whose top and bottom

are parallel and are circles. An example of a cylinder is a can of soup.

Example 30. sphere: A sphere is a round three dimensional figure such as a beach ball or a basketball. A sphere has a center and a radius like a circle, except it consists of all points in space which are the same distance from the center.


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Labeling Polygons

Use the letters on the vertices (plural of vertex) to label a polygon. For example, the polygon below could be labeled ABCDE or DEABC or EDCBA, etc. Always proceed around the polygon either clockwise or counterclockwise when labeling the vertices.

Note that a polygon can be very irregular. Two letters are often used to specify a certain side of a figure. The upper left side of the figure above is a line segment and could be

labeled AB . Angle Labeling and Measurement

Although single letters are generally used in geometry to designate single points, occasionally a single letter will denote an angle in a polygon. For example, the top angle in the figure above can be labeled just B with no chance of misunderstanding. It can also be labeled ABC or CBA. If there is danger of confusion, as in the figure below, you cannot label an angle with a single letter. You must use three letters with the vertex always the middle letter.


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AK, DK, CK, EK, and BK are all line segments and the corresponding rays are the

sides of various angles. For instance, the rays containing AK and DKare the sides of AKD. Note that using the single letter K for this angle would tell us nothing, since several angles have point K as their common vertex.

BKC is a right angle, because it contains the symbol at its vertex. Its measure is 900. If an angle does not contain this symbol, and you have not been told that its measure is 900, no matter how the picture looks, you cannot assume that the angle is a right angle. AKC must be a straight angle, because it consists of two right angles. Its measure is 900 + 900 or 1800. EKD is an obtuse angle, because it measures more than 900.

DKC is an acute angle, because it measures less than 900.

It can be shown by experiment or by mathematical proof that all of the measures of the angles in a triangle add up to 1800.

Example 31: Given the triangle below, what is the measure of each angle?

Answer: Because three angles of a triangle measure 180 , we can set up and solve an equation to determine the measure of each angle.

x 2x 3x 180

6x 180

x 30

If x 30 , then 2x 60 , and 3x 90 . Therefore, the angles of this triangle

are 30 , 60 , and 90 which sum to 180 .


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If you wish to know the actual measure of an angle, you may use a measuring device called a protractor. To read the measures of the angles using a protractor as shown in the diagram below, use the measures on the lower scale starting with 00 on the horizontal measuring from right to left.

Example 32: What is the measure of the angle below?

Answer: The measure of the angle is 30 degrees.


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Answer: The measure of the angle is 135 degrees.


Answer: The measure of the angle below is 45 degrees.

Any polygon can be divided into two fewer triangles than it has sides. In other words, a quadrilateral can be divided into two triangles, a pentagon can be divided into three triangles, etc. Three such divisions are shown.


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Using this result, it can be shown that the measures of the angles in any polygon total (n - 2)1800, where n is the number of sides in the polygon. Using this result, we find the measures of the angles in a quadrilateral total (4 - 2)(1800) or 3600; likewise, the measures of the angles in a pentagon total (5 - 2)(1800) or 5400, etc. Similar Polygons

When two polygons have exactly the same shape (not necessarily the same size), they are said to be similar. When two polygons are similar, then all of the angles of one polygon are the same measure as all of the angles of another polygon when read sequentially around the polygon. Marking Equal Angles

There are standard ways of marking angles whose measures are the same. The following two angles are marked with an arc to show that their measures are the same even though they do not lie in the same position:

The following two angles are marked differently to show that they have the same measure as each other but not the same measure as the previous two angles:

In other words, all angles marked with one type of arc are the same size; all angles marked with a different type of arc are the same size; etc. Other schemes for marking angles of the same measure are available. Angles marked the same way are the same size (have the same degree measurement).

Unit 4 Geometry and Units of Measurement Section 4.1 Problems

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1. Suppose BKE measures 1150. Find the measure of AKB in the figure on the right.

2. Name two acute angles in the

figure on the right. 3. Name two right angles in the figure

on the right. 4. Name two obtuse angles in the

figure on the right. 5. Find each angle of the triangle on

the right.

Hint: The measures of the angles total 1800.

6. Find the missing angles of the triangle on the right.

7. Find the missing angles of the triangle on the right.


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8. Identify a physical model for each of the following:

a) an acute angle b) a right angle c) a square

d) a triangle e) a parallelogram f) a rhombus

9. Name the following polygons (be as specific as possible):

10. Two angles are said to be complementary if the sum of their measures is 90 degrees. Suppose two angles are complementary and equal. Find the measure of each angle.

11. Using the figure below, can you find any complementary angles?

12. Two angles are said to be supplementary if the sum of their measures is 180 degrees. Using the figure above, can you find any supplementary angles?

13. Find the radius of a circle whose diameter is 46 cm.

14. If all three angles of a triangle are equal, what must be the size of each angle?

15. If all six angles of a hexagon are equal, what must be the size of each angle?


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16. What is wrong with the statement "A line is 6 inches long?”

17. If a round pizza is cut (through its center) into 8 equal pieces, what is the size of the angle at the tip of each piece?

18. If one angle of a triangle is 1000, and the other two angles are equal, how big is

each of the other two angles? 19. If a square is cut in half diagonally, how large are the angles of the resulting two

triangles? 20. What is the difference between a circle and a sphere? Do not use the word "like" in

your answer. 21. One angle of a parallelogram measures 700. Find the measures of the other three

angles. 22. What is the measure of the angles below?

a.


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b.

23. What are the measures of the angles of the triangle below?

24. What is wrong with the statement: “any line segment connecting two points on a circle is a diameter”?

Unit 4 Geometry and Units of Measurement Section 4.1 Answers to selected problems

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1. 250

2. BKA, CKD

3. BKC, AKE

4. EKD, AKC, EKC

5. 540, 580, 680

6. 800, 800

7. 330,1260,210

8. answers vary

9. a) hexagon b) rectangle c) triangle d) pentagon e) parallelogram f) octagon

10. 450

11. BKF and FKC

12. BKC, AKE

13. 23 cm

14. 600

15. 1200

16. Use "line segment" instead of line.

17. 450

18. 400

19. 450, 450, 900

20. Circle is two dimensional. Sphere is three dimensional.

21. 1100, 700, 1100

22. a. 600 b. 1500

23. 300, 450, 1050

24. The line segment must pass through the center of the circle.

Unit 4 Geometry and Units of Measurement Section 4.2 Triangles

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Objectives for this section Find missing sides in similar triangles Find missing sides in right triangles

Similar Triangles

The two triangles below are similar if we assume that the angle measurements are

equal, i.e., m A m D; m B m E; and m C m F . The m A means the

measure or size of A . We will usually omit the letter “m” and state simply

A B rather than m A m B .

When two triangles are similar, the ratios of each side of one of the triangles to the corresponding side of the other triangle are all the same. In other words, if one side of the larger triangle is twice as long as the corresponding side of the smaller triangle, then every side of the larger triangle is twice as long as the corresponding side of the smaller triangle. In general, two geometric figures are said to be similar if they have the same shape but not necessarily the same size. How these ratios may be used in computing missing sides for triangles is shown in the following example. Example 1. Suppose that the two triangles shown below are similar and that the

angles labeled with the same letter are the same size. Find the missing sides labeled x and y.


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Answer: First decide which sides of the large triangle correspond to which sides of the small triangle. The 40 m side of the large triangle connecting vertex A and vertex B corresponds to side y of the small triangle

connecting vertex a and vertex b. (Remember A and B are the same

size as a and b, respectively.) By the same reasoning, the 50 m side of the large triangle corresponds to side x of the small triangle, and the 20 m side of the large triangle corresponds to the 14 m side of the small triangle. We can now write a proportion with a pair of corresponding sides forming the top and the bottom of one ratio and another pair of corresponding sides forming the top and the bottom of another ratio. Both numerators must come from the same triangle, and both denominators must come from the same triangle. Hence, the three

ratios to be used are y x 14

, , and 40 50 20

. One side of the proportion must

contain an unknown, either x or y, and the other side must not contain an unknown. The correct proportion containing the unknown x is:

x 14

50 20

20x = (50)(14)

x = 35 m

Next, write a proportion to find the other unknown side.

y 14

40 20

Solving this proportion gives y = 28 m.


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Example 2. If a 5-foot-tall person casts a 12-foot shadow at a certain time of the day, and a tree casts a 66-foot shadow at the same time of the day, how tall is the tree?

Answer: The tree and its shadow and the person and the person’s shadow form

two triangles as shown.

The two triangles are similar. They both have right angles and the angle of elevation of the sun is the same in both. (Both measurements were taken at the same time of the day.) Write the proportion:

height of tree length of tree's shadow

height of person length of person's shadow

h 66

5 12

Solving for h gives: 12h = (5)(66)

h = 27.5 feet


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For any two similar triangles ABC and DEF, we have the following proportion:

first side of triangle ABC second side of triangle ABC

corresponding side of triangle DEF corresponding side of triangle DEF

Pythagorean Theorem

It is not necessary for triangles to be right triangles in order for them to be similar. However, the Pythagorean Theorem, discussed here, is true only for right triangles. A right triangle is one which has one right angle. The side of the triangle opposite the right angle is called the hypotenuse. The hypotenuse is the longest side of the triangle, and its length is often given by the letter c, as in the following figure. The lengths of the other two sides are often called a and b.

Example 3. Find the missing side of the triangle shown:

Answer: Since the triangle is a right triangle, we can use the Pythagorean Theorem. Since the missing side is the hypotenuse, we use the equation

2 2 230 72 x

2 230 72 x

78 feet = x


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Example 4. Find the missing side of the following triangle.

Answer: Since the triangle is a right triangle and the missing side is not the hypotenuse, we use the equation

2 2 2x 24 40

2 2 2x 40 24

2 2x 40 24

x = 32 m

Glossary

Right triangle-- a triangle that has one right angle (an angle equal to 900).

Hypotenuse--

the longest side, the one opposite the right angle.

Legs--

the two shorter sides, the ones forming the right angle.

Obtuse triangle--

a triangle that has one obtuse angle (an angle greater than 900).

Acute triangle-- a triangle that has three acute angles (angles less than 900).


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Equiangular triangle--

a triangle with three angles whose measures are each 600.

Scalene triangle— a triangle that has no sides of equal length.

Isosceles triangle-- a triangle with at least two sides of equal length.

Legs--the equal sides

Base--the remaining side

Base angles— angles on either side of the base and opposite the legs

Equilateral triangle-- a triangle that has three equal sides. For triangles, equal sides imply equal angles and vice-versa.


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1. If a tree casts a 20-meter shadow at the same time of day that a 2-meter tall person casts a 5-meter shadow, how tall is the tree?

2. If a person who is 75 inches tall casts a 21-foot shadow, how long a shadow will

a person who is 50 inches tall cast at the same time of day? 3. Find p and q (to the nearest whole number) in the following triangles given

that an angle marked with an upper case letter is the same size as an angle marked with the corresponding lower case letter.

4. Find x and y (to the nearest centimeter) in the following triangles, given that

the following pairs of angles are the same size: A and a, B and b, C

and c.

5. Find x and y to the nearest tenth of a foot in the following triangles, given that

A is the same size as a, B is the same size as b, and C is the same size

as c.


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6. Find the missing side (to the nearest tenth of an inch) of the right triangle shown here.

7. Find the missing side (to the nearest tenth of a meter) of the right triangle shown below.

8. Find the missing side (to the nearest tenth of a yard) of the right triangle shown below.

9. Find the missing side (to the nearest tenth of an inch) of the right triangle shown below.

10. A baseball diamond is 90-feet square, i.e., it is actually a 90 ft by 90 ft square. To the nearest tenth of a foot, how far is home plate from second base (the diagonal of the square)?


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11. A 25-ft ladder leans against a wall. If the bottom of the ladder is 5 ft from the wall, how far up the wall does the ladder reach? Round your answer to the nearest hundredth of a foot.

12. Can the sides of a right triangle measure 2 cm, 3 cm, and 4 cm?

13. Can the sides of a right triangle measure 21 cm, 28 cm, and 35 cm?

14. A football field is 360 feet long and 160 feet wide. What is the length of the diagonal? (to the nearest foot)

15. Libby walked three miles South and then four miles East. How far is she from

her starting point? 16. A twenty-five foot ramp covers twenty-four feet of ground. How high does the

ramp rise? 17. A rectangular sheet of paper is 8.5 in wide and 11 in long. Find to the nearest

tenth of an inch the length of the longest straight line segment which can be drawn on the paper.

18. Classify the following triangles as either isosceles, equilateral, or scalene:

a) b) c)

19. Classify the following triangles as either acute, obtuse, or right.

a) b) c)


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1. person

tree

5 2

20 x

x = 8 meters

2. 14 feet

3. 29 37

588 p

p = 750

29 19

588 p

q = 385

4. x = 520 cm y = 640 cm

5. 112 147

x 83

x = 63.2 ft

63 147

y 83

y = 35.6 ft

6. 48.8 in

7. 2 223 113 115.3 m

8. 211.5 yd

9. 2 266 36 55.3 in

10. 127.3 feet

11. 24.49 ft

12. No, since 2 2 22 3 4

13. Yes

14. 394 ft

15. 5 mi

16. 7 feet

17. 2 28.5 11 13.9 in

corner to corner 18. a) scalene

b) equilateral c) isosceles

19. a) obtuse b) right c) acute

Unit 4 Geometry and Units of Measurement Section 4.3 Length, metric system

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Objectives for this section Convert from one common metric unit of length to another Find the perimeter of figures

Linear measure and units used for this purpose

There are two systems used for measuring distances (called linear measure). They are the English system (sometimes called the customary system), used widely in the United States and very little elsewhere, and the metric system (or SI), used in many applications in the United States and almost exclusively in the rest of the world. English units for measuring distances are the inch, the foot, the yard, the mile, the furlong, the nautical mile, etc. These units have been around for a long time and have interesting histories, but the conversions from one unit to the other follow no pattern. There are 12 inches in a foot, 3 feet in a yard, 5,280 feet in a mile, and so on. The conversions are totally different for each unit. One reason for this chaos is that our measuring system was originally based on human measure, such as the span of some person's hand, a quantity which varies from person to person. A mile was 1000 steps of a Roman soldier. The word “mile” comes from the Latin word “mille” meaning one thousand. Was it a tall soldier or a short one? The yard was the length from the king’s nose to the end of his thumb with his arm outstretched. Long-nosed king or what? Obviously, the differences in body sizes frustrated creation of a uniform measuring system. So in 1791, French scientists created the metric system. Ten million meters were supposed to be the distance from the North Pole to the equator, and this distance was scratched on a metal rod. This distance turned out not to have been accurately measured, and the meter is now defined in terms of the wavelength of

some frequency of light. It is also the distance that light travels in 1

299,792,458 of a

second. The meter is the official distance in the United States, as it is in almost every other country in the world. However, the metric system is not used nearly as widely in the U.S. as it is in the rest of the world. The metric system attempts to remedy the problem of many unrelated conversions by defining just one unit of length, the meter, and making every other unit some power of ten times the basic unit. For instance, one kilometer is 1,000 meters. One

centimeter is 1

100 of a meter or 0.01 meter.


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The real advantage of this system is that conversions from one unit to the other are very easy. As a unit of length, the meter has less practical history, and is certainly no more useful than the foot. Typical commonly used units of length in the metric system would be the meter, the centimeter, the millimeter, and the kilometer. Since the metric system is based on ten (as is our entire numbering system), all units

are given as decimals, not common fractions. That is, 6.5 meters, not 1

62

meters, is

the correct notation. All common units of length in the metric system contain the word “meter,” together with a prefix which tells what multiple or fraction of a meter is being used. Some prefixes are shown in the following chart.

Prefix Abbreviation Power of 10 Meaning

exa E 1018 quintillion

peta P 1015 quadrillion

tera T 1012 trillion

Giga G 109 billion

Mega M 106 million

kilo k 103 or 1,000 thousand

hecto h 102 or 100 hundred

deka da 101 or 10 ten

no prefix 100 or 1 one

deci d 10-1 or 0.1 tenth

centi c 10-2 or 0.01 hundredth

milli m 10-3 or 0.001 thousandth

micro or mc 10-6 millionth

nano n 10-9 billionth

pico p 10-12 trillionth

femto f 10-15 quadrillionth

atto a 10-18 quintillionth

Hence, a kilometer is 1000 meters, a millimeter is one one-thousandth of a meter or 0.001 meter, a nanometer is one one-billionth of a meter or 0.000 000 001 meter, a megameter is one million meters, a centimeter is one one-hundredth of a meter or 0.01and so on. These prefixes can be used with any basic metric unit of measurement. If you are driving down the highway in a metric system country (or even some states in this country), the distance to the next town will most likely be measured in kilometers (abbreviated km). The kilometer is a little over a half mile, actually about 0.6 miles. So 100 km per hour is about the same as 60 miles per hour.


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Another commonly used metric measure of length is the meter itself (abbreviated m and sometimes spelled metre). The meter is a little longer than the yard. It contains about 39.37 inches, while the yard contains exactly 36 inches. A standard Olympic event is the 100-meter dash. Because the meter is a little longer than the yard, the 100-meter dash is longer than the 100-yard dash In applications where the inch is used in the English system, the centimeter (abbreviated cm) is generally used in the metric system. An inch is exactly 2.54 centimeters. Looking at it another way, a centimeter is a little less than a half inch. The width of a standard paperclip is about one centimeter. In a hospital using the metric system, the length of a newborn baby would probably be measured in cm (centimeters). The height of an adult can be measured in either meters or centimeters. For instance, a person who is 5 ft 8 in tall (68 in) would be (68)(2.54) or about 173 cm tall. The height could also be expressed as 1.73 m. The smallest commonly used metric unit of length is the millimeter. This unit is used for the size of tools needed to work on metric system machinery. Most pencil leads are no thicker than 1 millimeter (abbreviated mm); common sizes are 0.5 mm and 0.7 mm. The wire in an ordinary paper clip is about one mm thick. The smallest bird egg laid by the Vervain Hummingbird of Jamaica is 9.9 mm in length. The average diameter of a human hair is about 0.063 5 mm. Metric system countries measure the diameter of the bullet in a firearm in millimeters rather than in hundredths of an inch as we do. You may read that the military and many police departments use 9 mm pistols. This means that the diameter of the bullet that the pistol shoots is 9 mm. One millimeter is about 0.04 inches; so the diameter (or caliber) of a 9 mm bullet is actually about 9 times 0.04 = 0.36 inches. Conversions from one unit to another are very easy in the metric system. It is accomplished by moving the decimal to the right, if changing from a larger unit to a smaller unit, or to the left, if changing from a smaller unit to a larger unit. Remember: DOWNRIGHT That is, if the size of the unit is going DOWN, move the decimal to the RIGHT. The amount that the decimal is moved is the difference in the powers of ten associated with each unit.


320

For instance, to convert km to cm, note that the power of ten that goes with km is

310 while the power that goes with cm is 210 . The difference between the powers is 5, and the size of the unit is going down. Thus, you need to move the decimal 5 places to the right in order to convert from km to cm. Some rules for metric units:

1. Symbols never end with a period -- cm not cm. 2. There should be one space between the number and the symbol -- 6 cm, not

6cm. 3. Symbols are never written in plural form -- 6 cm, not 6 cms. 4. Do not use capital letters unless they are needed for metric symbols. For

example, liters uses the symbol L.

Example 1. Change 5 km to cm.

Answer: Move the decimal 5 places to the right to get 500,000 cm.

Example 2. Change 80,000 mm to m.

Answer: Move the decimal 3 places to the left to get 80 m.

Example 3. Change 0.06 dm to mm.

Answer: Move the decimal 2 places to the right to get 6 mm.

One way to remember the prefixes kilo, hecto, deka, unit without a prefix or base unit, deci, centi, and milli is “King Henry Died By Drinking Chocolate Milk.” The B is for the base unit of measurement (meter, liter, gram).

Perimeter

The perimeter of a region is the distance around its boundary, and is found by adding together the lengths of all of the sides of the region. Example 4. Find the perimeter of a 6 ft by 10 ft rectangle.

Answer: Since there are two sides which are 6 ft long and two sides which are 10 ft long, calculate (2)(6) + (2)(10) = 12 + 20 = 32 ft.


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Notice that the units for a perimeter are the same as the units for a length. Additionally, all sides must be measured in the same units in order to calculate the perimeter of a figure. Example 5. Calculate the perimeter of a triangle whose sides are 5 m, 300 cm, and

70 dm. Answer: Change all measurements to the same units. In this case we will use

centimeters. Then add them together to obtain 500 cm + 300 cm + 700 cm = 1500 cm. You may then express the answer in any unit you wish; for example, 15 m.

For circles perimeter is called circumference, and must be calculated using a special

formula C d, where d is the diameter of the circle and is a non-ending, non-

repeating decimal equal to approximately 3.14159265359. Often is approximated

by 3.14 or by 22

7. However, the most accurate way to calculate using is to use the

key on your calculator instead of the approximations. Example 6. Find the circumference of the circle below:

Answer: Note that the radius of the circle is 60 cm. Since the diameter is twice the radius, the diameter is 120 cm. Calculating the circumference,

C (120) 376.9911184 or approximately 377 cm.

Note that the circumference of a circle will always be in the same units as the radius or the diameter used to calculate it. For many calculations 3.14 is accurate enough

to use as an approximation for . However, for the purposes of this course, we will

use the key on our calculators.


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The perimeter of some figures may require a combination of formulas to compute.

Example 7. Find the perimeter of the figure shown below, assuming that the right end is a quarter-circle.

Answer: First find the length of the quarter-circle. The radius is obviously 80 cm (Why?). Hence, the diameter is 160 cm, and the circumference is

(160) 502.6548246 cm . Multiplying by 0.25, since the end is only a

quarter-circle (one-fourth of a circle), gives 125.6637061 cm.

Then add 150 + 80 + 70 + 125.6637061 to get 426 cm rounded to the nearest whole cm.

Example 8. Use algebra to find the radius of a circle whose circumference is 1800 cm.

Answer: First find the diameter by using the formula for the circumference,

C d. Then solve for d. Thus,

Cd , which gives

1800572.9577951 cm for the diameter. The radius is half of the

diameter, or 286 cm rounded to the nearest whole cm.


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Example 9. Find the distance around the semi-circle shown below.

Answer: Note that the radius of the semi-circle is 70 m. The bottom side is the diameter of the semi-circle which is (2)(70m) or 140 m. The length of the curved portion is half of the circumference of a whole circle which is

(0.5)( )(140) 219.9114858 . The total distance around the figure

(perimeter) is 140 + 219.9114858, which is approximately 360 m when rounded to the nearest whole m.

Example 10. Find the perimeter of the right triangle shown below:

Answer: First find the hypotenuse by using the Pythagorean Theorem.

2 210 24 26 in

Then add all three sides together, 10 + 24 + 26 = 60 in, to find the perimeter.


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1. Find the perimeter of a rectangle which is 44 ft x 96 ft.

2. Find the perimeter of a rhombus which is 30 meters on each side.

3. Find the perimeter of a regular hexagon which is 50 yd on each side.

4. Find the perimeter of a regular octagon which is 20 in on each side.

5. The perimeter of a football field in the NFL is 2

3063

yards. How wide is the

field? Assume the length is 100 yards. 6. If the circumference of the earth is 25,000 miles, what is the distance from the

surface of the earth to its center? 7. The length of a rectangle is one centimeter more than twice the width. The

perimeter is 110 centimeters. Find the length and the width of the rectangle. 8. A square has a perimeter of 58 inches. What is the length of each side? 9. You are going to enclose your rectangular backyard with a fence. You will

fence only three sides - your house will be the fourth side. The back of your house is 56 feet long, and the distance from the back of your house to the rear property line is 108 feet. How many feet of fencing should you buy?

10. Find the circumference of a circular flower bed if the diameter is 10 meters.

11. Find the perimeter of a square whose side measures 0.8 meters.

12. Choose an appropriate metric unit to measure each of the following:

a) length of a new pencil

b) diameter of a nickel

c) width of your textbook

d) height of the doorknob from the floor

e) diameter of a piece of your hair


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f) height of a door

g) thickness of your textbook

h) length of an 18-wheeler truck

i) length of your bath tub

13. If the radius of a wheel is 30 cm, find its circumference.

14. If one mile is approximately 1.61 kilometers, and the distance to Los Angeles from New York is 4,690 km, how many miles is it?

15. If the distance from Miami to Atlanta is 642 miles, how many kilometers is it?

(see previous problem) 16. The speed limit on a certain section of I-85 in Spartanburg is 65 mph. Find this

speed in kilometers per hour. 17. Circle the larger measurement.

1 inch or 1 centimeter

1 yard or 1 meter

1 meter or 1 kilometer

6 inches or 13 centimeters

6 feet or 2 meters

1 foot or 35 centimeters

18. Complete each of the following:

a) 17.2 mm = ________ cm

b) 23.0 cm = ________ m

c) 262 m = ________ hm

d) 0.531mm = ________ dm

e) 0.17 km = ________ m


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19. Find the perimeter of this figure in cm to the nearest hundredth of a centimeter.

20. Find the perimeter of this figure to the nearest hundredth of a centimeter. The left end is a half-circle. The upper right corner is a right angle. The top and the bottom are parallel.

21. A surgeon wishes to take a 4-in long by 2-in wide skin graft from a donor site. How long will this graft be in cm? How wide? Do not round your answer.


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1. 44 + 44 + 96 + 96 = 280 ft

2. (4)(30) = 120 meters

3. (6)(50) = 300 yards

4. (8)(20) = 160 in

5. 1

533

yd

6. 3979 miles rounded to the nearest mile

7. width = 18 cm length = 37 cm

8. 14.5 in

9. 272 ft

10. 31.4 m rounded to the nearest tenth of a meter

11. 3.2 m

12. a) cm b) mm or cm c) cm d) cm or m e) mm

f) m or cm g) cm or mm h) m i) m or cm

13. 188.5 cm rounded to the nearest tenth of a cm

14. 2913 miles rounded to the nearest mile

15. 1034 km rounded to the nearest km

16. km

104 hr

17. a) 1 inch b) 1 meter c) 1 kilometer

d) 6 inches e) 2 meters f) 35 centimeters


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18. a) 1.72 b) 0.230 c) 2.62

d) 0.00531 e) 170

19. Change all dimensions to cm as shown.

Then use the Pythagorean Theorem to find the sloping sides.

2 2 2 228 25 37.53664876 and 17 25 30.23243292

The perimeter of the figure rounded to the nearest hundredth is:

28.00 40.00 17.00 30.23243292 40.00 37.53664876 192.77 cm

20. Find the circumference of a half-circle.

1

60.00 94.247779612

Find the length of the sloping edge using the Pythagorean Theorem.

2 2

90.00 70.00 60.00 25.00 40.31128874

Adding all of the dimensions together and rounding to the nearest hundredth, we get:

94.24777961 90.00 25.00 40.31128874 70.00 319.56 cm

21. 10.16 cm long; 5.08 cm wide

Unit 4 Geometry and Units of Measurement Section 4.4 Area

329

Objectives for this section Calculate areas of common geometric shapes and combinations Convert from one common metric unit of area to another

Area Measurement and Computation

The area of a plane (flat) region is measured in square units. For instance, the area of a 6 cm by 6 cm square would be (6 cm)(6 cm) = 36 cm2. You may say either “square centimeters” or “centimeters squared,” but you should always write “cm2” in the metric system not “sq cm.” In order to find the area of a figure, all dimensions must be given in the same units. For instance, if a rectangle measures 3 ft by 40 in, we could convert this to 36 in by 40 in and multiply (36 in)(40 in) to get an answer of 1440 in2 or 1440 sq in. Use of “sq in,” “sq yd,” etc. is OK in the English system. It does not matter which unit you choose to use for the dimensions, but the units must be the same if we are calculating perimeter, area, etc. The area of any figure always comes from multiplying together two distances.

Area of a Parallelogram

The area of any parallelogram is the product of its base (b) and its height (h).

A=b h

The height, h, of any figure is the vertical distance from its base to the opposite side or to a vertex. That is, the height of the parallelogram must be measured along a line perpendicular to the base. The lengths of the slanting sides are not used in the computation of the area. This formula also applies to rectangles and squares, since they are just special cases of parallelograms.


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Example 1. Find the area of this square.

Answer: Since the figure is a square, its base and its height are the same. Therefore, the area is (50 cm)(50 cm) = 2500 cm2.

Note: In ordinary language there is a difference in the statement “9 square

feet” and the statement “9 feet square.” The statement “9 square feet” means that the area of the figure is 9 ft2. The figure could be a square measuring 3 ft by 3 ft or a rectangle measuring 1ft by 9 ft. The statement “9 feet square” refers to a square whose dimensions are 9 ft by 9 ft. Thus, the actual area of such a square is 81 ft2.

Area of a Trapezoid

A trapezoid has only one pair of parallel sides. We call these two sides the bases of the trapezoid. The area of any trapezoid is the product of the average of the long base and the short base multiplied by the height.

1 2b b

A h2

Once again, the lengths of the slanting sides are not used in the computation of the area.


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Example 2. Find the area of the trapezoid shown below:

Answer:

30 cm 44 cm 74 cmA= 12 cm 12 cm 37 cm 12 cm

2 2444 cm2

rounded to the nearest whole cm2.

Area of a Triangle

Since a triangle is just half of a parallelogram, the formula for its area is

1

A b h2

Since the height of a triangle is the perpendicular distance from the base to the opposite vertex, you may have to venture outside of the triangle to find the height as shown in the following example:


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Example 3. Find the area of this triangle.

Answer: The area is 1

A 86 mm 67 mm2

2881 mm2.

Area of a Right Triangle

Because one leg (not the hypotenuse) of a right triangle is perpendicular to the other leg, one leg may be used as the base and the other leg used as the height of the triangle as shown in the following example.


Answer: The area is 1

A 22 mm 37 mm2

407 cm2.


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Answer: Use the Pythagorean Theorem to find the length of the missing leg which

is 9 in. The area is 1

A 9 in 12 in2

54 in2.

Example 6. Find the area of the figure enclosed by the heavy solid line. Often,

more complicated shapes can be broken up into triangles and rectangles, and their areas computed by adding or subtracting simpler areas as illustrated by the figure below. The dotted lines have been drawn to show how to break the figure into simpler shapes.


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Answer: The strategy for finding the area of this figure is to first find the area of the large rectangle at the bottom. Next, we subtract the area of the corner cut off the bottom left. Note that you must subtract to find the base and the height of the small triangle. Finally, we add the area of the large triangle on top.

Area = Area of bottom rectangle - Area of small bottom left triangle which has been cut off + Area of large triangle on top

1 1

A 40 cm 46 cm 46 cm 38 cm 40 cm 30cm 46 cm 30 cm2 2

= 1840 cm2 – 40 cm2 + 690 cm2 = 2490 cm2

Don’t forget to include units with your answers.

Another common figure for which there is a known area formula is the circle.

Area of a Circle

2A r where r is the radius of the circle

Example 7. Find the area of the circle shown rounded to the nearest whole cm2.

Answer: Note that the dimension shown is the diameter, not the radius. The radius is half of the diameter or 20 cm. Hence, the area is

2 2 2A 20 cm 400 cm 1256.637061 cm 1257 cm2.


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Conversion from one Area Unit to Another

One metric area unit may be converted to another using the method shown in the following examples: Example 8. Convert 4 cm2 to mm2

Answer: 4 cm2 = (4)(1 cm)(1 cm) = (4)(10 mm)(10 mm) = 400 mm2

Example 9. Convert 5 yd2 to in2.

Answer: 5 yd2 = (5)(1 yd)(1 yd) = (5)(36 in)(36 in) = 6480 in2

Example 10. Convert 6000 m2 to km2.

Answer: 6000 m2 = (6000)(1 m)(1 m) = (6000)(0.001 km)(0.001 km) = 0.006 km2

Area is not always measured in square units. We use the acre to measure the area of land in the English system of measurement. The acre is 43,560 ft2. There is a similar metric unit called the hectare which is 10,000 m2; that is, a square that measures 100 meters on a side. It would be difficult to start using the hectare for land measurement in the United States. Why? Think of the millions of property descriptions in courthouses all over the country. Virtually every one of them gives the size of land in acres. Since land isn’t moveable and can’t be exported, there is little advantage to measuring land area in a metric unit.


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1. Find the area of a square which has a side of 60 mm.

2. Find the area of a right triangle whose legs are 20 in and 30 in.

3. Find the area of a rectangle which is 4 ft wide and 6 in high.

4. Find the area of a circle whose diameter is 16 inches rounded to the nearest in2.

5. Find how many square feet of tile must be purchased to tile the patio shown

below if you must buy 5% extra to allow for waste. The patio is a half-circle with a rectangle cut out. Give your answer to the nearest ft2.

6. Find the area of the figure below. The upper left corner is one-fourth of a circle. The dotted lines should help you decide how to divide the area into simpler figures. Give your answer to the nearest cm2.


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7. Find the area of the unshaded part of the circle. Give your answer to the nearest cm2.

8. Find the area of the right triangle shown below:

9. What is the difference between 2 m square and 2 m2?

10. If the ratio of the sides of two squares is 2 to 3, find the ratio of their areas.


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11. Find the area of the large room whose floor plan is shown below. Round off to

the nearest tenth of a square meter. Note that the figure is a trapezoid with a small rectangle cut from the bottom.

12. Find the area of the figure shown (three-quarters of a circle and a small square). Round the answer to the nearest square inch.

13. Find the perimeter of the figure above to the nearest tenth of an inch.

14. One mile is approximately 1.61 kilometers. Find approximately how many km2 are in one square mile rounded to two decimal places.

15. A worker uses stick-on carpet squares that measure 10 cm by 10 cm to cover

his bathroom floor. If the bathroom measures 4 meters by 5 meters, find how many tiles the worker will need to carpet the room.

16. If a counter top measures 2 ft by 6 ft, find how many 4 in by 4 in ceramic tiles

it will take to cover the countertop if you must buy 5% extra to allow for waste.


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17. If a circle has a circumference of 12 meters, find its area to the nearest m2.

18. A circular flower bed is eight meters in diameter and has a circular sidewalk around the outside of it. The sidewalk is one meter wide. Find the area of the sidewalk rounded to the nearest square meter.

19. Find the area of a parallelogram whose base is fourteen centimeters and whose

height is six centimeters. 20. A ceiling measuring 9 feet by 15 feet can be painted for $60.00. Find the cost

to paint a ceiling measuring 18 feet by 30 feet. 21. Find the area of a right triangle whose hypotenuse measures 10 feet and whose

base measures 8 feet. 22. Which size pizza is the best buy (the least money per square inch of pizza)?

The pizzas are round.

10 in for $5.99

12 in for $7.99

14 in for $8.99

23. Find the area of a rectangular home with dimensions 60 feet by 28 feet.

24. Circle the larger measurement in each comparison:

a) 1 square inch or 1 square cm

b) 1 square foot or 1 square meter

c) 1 square mile or 1 square km

d) 1 square yard or 1 square meter

25. A rectangular piece of land measures 1300 meters by 1500 meters. Find the area of this piece of land in square kilometers.

26. A rectangular piece of land measures 1300 yards by 1500 yards. Find the area

of this piece of land in square miles rounded to two decimal places.


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27. A square measures 20 cm on a side, and a circle has a diameter of 20 cm.

a) The area of the circle is what percent of the area of the square? Round your answer to the nearest tenth of a percent.

b) The circumference of the circle is what percent of the perimeter of the

square? Round your answer to the nearest tenth of a percent. Writing exercises

28. In order to purchase grass seed for a lawn, would you need the perimeter of the lawn or the area of the lawn? Explain.

29. In order to purchase fencing for a yard, would you need the perimeter of the

yard or the area of the yard? Explain.


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1. 3600 mm2

2. 300 in2

3. 2 ft2

4. 201 in2

5. 373 ft2

6. 4807 cm2

7. 4527 cm2

8. 5400 cm2

9. 2 m square is an old term for 2 m by 2 m which means an area of 4 m2

10. 4 to 9

11. 116.5 m2

12. 215 in2

13. 53.7 in

14. 2.59

15. 2000

16. First calculate the area of the countertop in square inches. (24 in)(72 in) = 1728 in2. Next, calculate the area of each tile in square inches. (4 in)(4 in) = 16 in2. Divide 1728 by 16 to find the number of tiles needed. The answer is 108 tiles. However, this figure must be increased by 5% to allow for waste.

Hence, the final answer is 108(1.05) = 114 tiles. In a situation like this you should always round the answer upward regardless of the fraction to ensure that you have enough material for the job.

17. 113 m2

18. 28 m2

19. 84 cm2

20. $240

21. 24 ft2

22. 14 in

23. 1680 sq ft

24. a) 1 sq in b) 1 m2 c) 1 sq mi d) 1 m2

25. 1.95 km2

26. 0.63 sq mi

27. a) 78.5% b) 78.5%

Unit 4 Geometry and Units of Measurement Section 4.5 Volume and capacity

343

Objectives for this section Calculate volumes of common solids and combinations Change metric volume units into metric capacity units and vice versa

Volume vs. Capacity

The volume of an object is the amount of space which it occupies. Since computing the volume of an object involves multiplying together three units of length, the unit of volume is always a cubic unit such as m3 or ft3 (often written cu ft), depending on whether the computations are being done in the metric or the English system. All of the dimensions used in computation of the volume of an object must be given in the same units. If we wish to find the volume of a rectangular box measuring 2 ft by 1 yd by 10 in, one possibility would be to change all dimensions to inches and to multiply (24 in)(36 in)(10 in) to get the volume in cubic inches (in3 or cu in). For many purposes, cubic units are not used to measure volume. Instead, capacity units are used. This is true both in the metric and in the English systems. Some standard units of capacity in the English system are the fluid ounce, the pint, the quart, the gallon, the bushel, the barrel, etc. There is only one standard unit of capacity in the metric system, the liter. The liter is one dm3 (one cubic decimeter). The liter is slightly more than the quart. Most capacities in the metric system are measured either in liters or in milliliters. To avoid confusion when using abbreviations, always capitalize (L) or write in script (l) the L which stands for liter. That is, write L (or l) for liter, mL (or ml) for milliliter, or dL (or dl) for deciliter. You should learn the following equivalencies:

The internal size of an engine in a car, a motorcycle, or other motorized vehicle is often measured in cubic inches in this country. However, as we become more metric, we are starting to use metric units most of the time. Hence, an automobile company will advertise their 5.7 L engine rather than their 350 cubic inch engine, even though the two measurements are about the same. The first is metric, and the second is English. The size of small engines such as those on chain saws, motorcycles, etc. are often measured in cc’s (same as cm3 or mL) instead of liters. A 900 cc engine would be the same as a 0.9 L engine, since there are 1000 cc (or 1000 mL) in 1 L. If you were to shop in a grocery store using the metric system, you would buy liquids such as milk, juices, soft drinks, etc. by the liter rather than by the quart or by the gallon. You would buy gasoline, motor oil, engine coolant, and most all liquid products by the liter rather than by the pint, by the quart, or by the gallon.


344

Volume of a Prism or a Cylinder

Calculating a volume always involves multiplying three lengths together. A prism is a solid figure whose top surface is parallel to the bottom surface, and whose top surface and bottom surface are the same size and shape. The sides of the prism are parallelograms. A cylinder has a circle for the top and bottom that are parallel to each other and the same size. The formula for the volume of a prism or cylinder is:

V B H

where B is the area of the base (bottom surface), and H is the height of the figure.

The height of a solid figure is the perpendicular distance from the bottom surface to the top surface. Example 1. Find the volume of the figure shown below assuming that the base is a

rectangle: Round your answer to the nearest in3.

Answer: First find the area of the base by multiplying (14 in)(16 in) to get 224 in2. Then multiply this by the vertical height which is 30 in (not 34 in). Therefore, the volume is (224 in2)(30 in) = 6720 in3.


345

Example 2. Find the volume of the solid figure below, assuming that the top and the bottom are circles and that they are parallel to each other. Round your answer to the nearest cm3.

Answer: Find the area of the circular base by multiplying 2(30 cm) to get

2827.433388 cm2 (use 30 for the radius since it is half of the diameter). This is the area of the base. Then multiply this by the vertical height of the object which is 50 cm (not 55cm). This gives 141,372 cm3 as the volume of the object. The answer can also be written 141,372 cc or 141,372 mL or 141.372 L since those are equivalent units.


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Volume of a Pyramid or Cone (pointed at top):

1

V B H3

Again, B is the area of the base.

Example 3. Find the volume of the pyramid shown below, assuming that the base is a 60 cm by 60 cm square:

Answer: First find the area of the base by multiplying (60 cm)(60 cm) to get 3600

cm2. Then multiply by the height which is 80 cm (not 110 cm), since the height must be measured perpendicular to the base. This gives (3600 cm2)(80 cm) = 288,000 cm3 which would be the correct answer if

the figure were not pointed at the top. Multiplying by 1

3 gives

31288,000 cm

396,000 cm3.


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Volume of a Sphere

3

34V r where r is the radius

Example 4. Find the volume of the sphere shown rounded to the nearest whole cm3.

Answer: Since the diameter is 60 cm, the radius is 30 cm. The volume is

34

30 cm3

113,097 cm3.


348

Combinations

Example 5. Find the volume of the figure shown below rounded to the nearest whole unit:

Answer: We must find the volume of the cone and add it to the volume of the

hemisphere (1

2 of a sphere).

Cone:

21 1V B H= r H

3 3

2 31V 4.00 cm 9.00 cm 150.7964474 cm

3

Hemisphere:

31 4V r

2 3

31 4V 4.00 cm 134.0412864

2 3

Total = 150.7964474 + 134.0412866 = 285 cm3


349

Application of Volume

The air in a room and the liquid in a container both involve volume measurement. The air conditioning maintenance engineer makes calculations to determine the rate at which air is replaced in any given area of a hospital. Example 6. If a patient’s room measures 5.750 m by 6.125 m by 3.000 m and a fan

can exchange 12.750 m3 per hour, how much time is required to exchange all the air in the room?

Answer: The volume of the room would be given by: V=B H, where B is the area of the base (a rectangle).

3V 5.750 m (6.125 m)(3.000 m) 105.65625 m .

Next, divide the volume by 12.750 m3 per hour

3

3

105.65625 m

m12.750

hr

8.287 hours

Changing from one Volume Unit to Another

To change a volume from one unit to another, proceed as shown in the following three examples: Example 7. Change 3 km3 into m3.

Answer: 3 km3 = 3(1 km)(1 km)(1 km) = (3)(1000 m)(1000 m)(1000 m) = 3,000,000,000 m3 Example 8. Change 200 mm3 into dm3.

Answer: 200 mm3 = 200(1 mm)(1 mm)(1 mm) = 200(0.01 dm)(0.01 dm)(0.01 dm) = 0.000 2 dm3

Example 9. Change 5 ft3 into in3.

Answer: 5 ft3 = 5(1 ft)(1 ft)(1 ft) = 5(12 in)(12 in)(12 in) = 8640 in3


350

1. How many cubic inches are there in a cubic yard?

2. How many cubic feet are there in a cubic yard?

3. Express the volume in scientific notation of a rectangular box which measures

width: 33.1 10 cm length: 21.5 10 cm height: 32.0 10 cm

4. If a coffee can has a radius of 63.0 cm and a height of 15.8 cm, find its volume rounded to the nearest cm3.

5. Find the volume of a bottle of typewriter correction fluid with a diameter of

3.0 cm and a height of 4.3 cm rounded to one decimal place. 6. Two cubes have sides of length 4.0 cm and 6.0 cm, respectively. Find the ratio

of their volumes. 7. A bread pan measures 18 cm by 18 cm by 5 cm. How many liters does it hold? 8. A standard straw is 25 centimeters long and 4 mm in diameter. How much

liquid can be held in the straw at one time? Round your answer to two decimal places.

9. A regular square pyramid is 3 meters high, and the perimeter of the base is 16

meters. Find the volume of the pyramid. 10. Convert each of the following:

a) 27 L = ______ mL b) 568 mL = _______ L

c) 5.0 mL = ______ cm3 d) 8.00 m3 = _______ L

11. The volume of a rectangular solid is 30 dm3. Find the volume in cubic centimeters.


351

12. Find the volume rounded to the nearest cubic millimeters:

a)

b)

13. Find the volume in cubic meters:

a)

b)


352

14. Find the capacity in liters rounded to two decimal places:

a)

b)

15. Find the volume of each of the following rounded to the nearest whole unit:

a)

b)


353

c)

e)

d)


354

16. Find the volumes of the following three-dimensional figures rounded to the nearest tenth.

a) The ends of the top are half circles.

b)


355

c) Building with modified mansard roof.

d) Cylinder with a hemisphere at each end.


356

1. 46,656 2. 27

3. 89.3 10 cm3 4. 197,010 cm3 5. 30.4 cm3 6. 43 to 63 or 64 to 216 7. 1.62 8. 3.14 mL 9. 16 m3 10. a) 27,000

b) 0.568 c) 5.0 d) 8000

11. 30,000 cm3 12. a) 75,398 mm3

b) 141,371,669 mm3 13. a) 0.00225 m3

b) 0.006 m3 14. a) 0.15 L

b) 0.42 L 15. a) 66 cm3

b) 180 cm3 c) 64 cm3 d) 4,189 cm3 e) 14 cm3

16. a)

2 3 31V B H 5 10 8 30 m 3578.097245 m

2

which rounds to 3578.1 m3.

b) 1

V B H B H3

2 2 3 3130 50 30 80 cm 273,318.5609 cm

3

which rounds to 273,318.6 cm3.

c) V B H

3 31 16 2516 5 7 25 14 60 ft 32,010 ft

2 2

which rounds to 32,010.0 ft3

d) 34V r B H

3

3 2 3 342 2 6 m 108.9085453 m

3

which rounds to 108.9 m3

Unit 4 Geometry and Units of Measurement Section 4.6 Miscellaneous metric units and summary

357

Objectives for this section Identify metric and English units of mass, temperature, length, area, volume, capacity Convert between Celsius and Fahrenheit temperature Select appropriate metric unit for various purposes Do English-metric conversions when conversion factor is given

Mass or Weight

The metric unit of mass is the gram. It is a very small unit, about the mass of a standard size paperclip and is abbreviated g or occasionally gm. One ounce in the English system is equivalent to about 30 grams in the metric system. A very small unit, the milligram (one thousandth of a gram), is used extensively in the medical and scientific fields. Often bottles of medicine are labeled using the unit mg (milligram or 0.001 g). Scientists distinguish between mass and weight. An object has the same mass no matter where in the universe it is located; whereas, its weight depends on whether it is located on the moon or on the earth. A person weighs less on the moon, but still has the same mass. The metric unit called the gram is actually a unit of mass, while the English unit called the pound is actually a unit of weight. We shall not concern ourselves about this distinction in this course. The kilogram is the metric unit of mass used in everyday applications, where the pound would be used in the English system. The kg is about 2.205 pounds. There is even a metric tonne (note the spelling) which is equal to 1000 kg or about 2200 pounds. In the English system, the customary ton equals 2000 pounds, although what is called a long ton equals about 2200 pounds. Computer Memory

The amount of memory in a computer is measured in units called bytes. Computers have memories which consist of thousands, millions, or even billions of bytes. Hence, in advertisements you will see the terms kilobytes, megabytes, and gigabytes referring to memory for computers. One gigabyte is 1,000,000,000 bytes! Future computers will probably use even larger units to measure memory. Time

The length of time it takes a computer to perform an operation is usually measured in billionths of a second. Hence you may order a computer chip which can perform an operation in 40 nanoseconds.


358

Electrical Units

The unit of electrical power is called the Watt. You may read that the local power company has a generating facility with a capacity of 50 million Watts or 50 megawatts or 50 MW. For comparison, a normal light bulb seldom uses more than 100 Watts. A hot water heater may use 5000 Watts or 5 kW. During one very hot period, power usage in this area was described in Gigawatts, or billions of Watts! An overhead power line may be a 50 KV line. This means that the line carries electricity with a strength of 50,000 volts. Temperature

The unit of temperature in the metric system is not the same as the unit of temperature in the English system. In the English system, 0o F and 100o F represent the extremes between a cold day and a hot day. In the metric system 0o C and 100o C represent the freezing point of water and the boiling point of water, respectively. We use degrees Fahrenheit (abbreviated F) in the English system and degrees Celsius (abbreviated C) in the metric system.

To convert from one temperature scale to another: 1. Add 40. 2. Divide by 1.8 (F to C) or multiply by 1.8 (C to F) 3. Subtract 40.

The two scales coincide at -400.

Some common temperatures given in both Celsius and Fahrenheit degrees

Application of temperature Fahrenheit temperature Celsius temperature

Boiling Point of Water 212 0F 100 0C

Hot Day 100 0F 37.8 0C

“Normal” Body Temperature 98.6 0F 37 0C

The 98.60 figure for normal oral body temperature in the English system is not actually accurate according to a 1992 report in the Journal of the American Medical Association. The number 98.6 was obtained by converting 37.00 C from Celsius into Fahrenheit degrees. Normal temperature was originally obtained by a German physician (Carl Reinhold August Wunderlich) in 1868 who used measurements in Celsius degrees. More recent measurements have indicated that the normal oral body temperature is closer to 98.20 F.

Comfortable Day 68 20 0C

Freezing Point of Water 320 F 0 0C

Cold Day 0 0C -17.8 0C

Two Temperature Scales Equal -40 0F -40 0C

Absolute zero (all motion stops) -459 0F -273 0C


359

TYPICAL UNITS OF MEASUREMENT IN EACH SYSTEM (not equal)

English Metric

Length

in (inch) mm (millimeter)

ft (foot) cm (centimeter)

yd (yard) m (meter)

mi (mile) km (kilometer)

Area

sq in or in2 (square inch) mm2 (square millimeter)

sq ft or ft2 (square foot) cm2 (square centimeter)

sq yd or yd2 (square yard) m2 (square meter)

sq mi or mi2 (square mile) km2 (square kilometer)

acre hectare

Volume

cu in or in3 (cubic inch) mm3 (cubic millimeter)

cu ft or ft3 (cubic feet) cm3 (cubic centimeter = cc = mL; weighs 1g if substance is water)

cu yd or yd3 (cubic yard) m3 (cubic meter)

cu mi or mi3 (cubic mile) km3 (cubic kilometer)

Note that units of area and volume in the metric system are always written with an exponent rather than with the abbreviation “sq” or “cu.” In other words, either “sq ft” or “ft2” is OK, but always use “m2” instead of “sq m.”

Capacity

minim mL (milliliter = cm3 )

teaspoon cL (centiliter)

tablespoon dL (deciliter)

fl dr (fluid dram) L (1 liter = 1dm3)

fl oz (fluid ounce) daL (dekaliter)

cup hL (hectoliter)

pt (pint) kL (kiloliter)

qt (quart) Blank

gal (gallon) Blank

bbl (barrel) Blank

bu (bushel) Blank

Weight or Mass

gr (grain) mg (milligram)

dr (dram) g (gram)

oz (ounce) kg (kilogram)

lb (pound) metric tonne

ton Blank

Temperature oF (degrees Fahrenheit) 0C (degrees Celsius--at one time called

Centigrade)

Blank 0K (degrees Kelvin--used in chemistry and physics)


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Useful equivalents

1 cc = 1 cm3 = 1 mL, and 1 mL of water has a mass of 1 g (know this)!

1 dm3 = 1 L, and 1 L of water has a mass of 1 kg

Unit analysis

Unit analysis is a mathematical way of changing from one unit to another and checking the results of calculations involving units of measurement. Units may be treated just like numbers or variables in an expression. That is, they may be added, subtracted, multiplied, divided, squared, cubed, etc. Example 1. Changing 2 yards to 6 feet involves another bit of unit analysis. How is it

done? Answer: Obviously, we multiplied (2)(3) to get 6. But why? Why not divide,

2 3, instead? If we show all units in the calculation, it becomes clear why we multiply rather than divide.

The calculation with the units shown becomes

3 feet

2 yards 1 yard

6 feet

“yard” on the top cancels out “yard” on the bottom

Example 2. Convert 48 inches to feet given the conversion 1 foot = 12 inches.

Answer: 1 ft

48 in 12 in

4 ft

"in" on the top cancels out “in” on the bottom

From this example it can be seen that, when changing from one unit to another, the idea of unit analysis is to always arrange the conversion so that the undesired unit (inches in this case) cancels out when the multiplication is done.


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Example 3. Convert 5 yards to feet given the conversion 1 yard = 3 feet.

Answer: 3 ft

5 yd 3 5 ft1 yd

15 ft

Conversions between the English and the Metric systems may be done using this method. Example 4. Convert 80.0 pounds to kilograms given the approximate conversion 1 kg

= 2.205 lb.

Answer: 1 kg 80.0

80.0 lb kg2.205 lb 2.205

36.3 kg

Conversions within the English system (exact)

12 in (inches) 1 ft (foot)

3 ft (feet) 1 yd (yard)

5,280 ft (feet) 1 mi (mile)

3 tsp (teaspoons) 1 tbsp (tablespoon)

2 tbsp (tablespoons) 1 fl oz (fluid ounce)

16 tbsp (tablespoons)

1 c (cup)

16 fl oz (fluid ounces)

1 pt (pint)

2 c (cups) 1 pt (pint)

2 pt (pints) 1 qt (quart)

4 qt (quarts) 1 gal (gallon)

16 oz (ounces) 1 lb (pound)

2,000 pounds 1 ton

Conversions between the English and the Metric systems (approximate)

1 in (inch) (exact conversion)

2.54 cm (centimeters)

1 mi (mile) 1.61 km (kilometers)

1 tsp (teaspoon) 4.93 mL (milliliters)

1 pt (pint) 473 mL (milliliters)

1 qt (quart) 0.946 L (liters)

1 oz (ounce) 28.3 g (grams)

1 lb (pound)

2.205 (pounds)

0.454 kg (kilograms) 1 kg

Example 5. Add 4 feet + 2 yards. When adding lengths (such as finding the perimeter), all lengths must be expressed in the same units so that the resulting like terms may be added together.

Answer: 4 feet + 6 feet = 10 feet. The yards were multiplied by 3 and changed

to feet. Then 4 feet and 6 feet are like terms just like 4x and 6x, and may be added.


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Example 6. Multiply (5 cm)(6 cm).

Answer: Multiply (5)(6) and also multiply (cm)(cm) to obtain 30 cm2.

Example 7. Multiply (3 yd)(4 yd)(5 yd).

Answer: Multiply (3)(4)(5) and also multiply (yd)(yd)yd) to obtain 60 yd3.

Unit analysis for conversions of area and volume units

The conversion for a square unit may be obtained by squaring the conversion for a unit of length. For instance, since 3 ft = 1 yd, then 9 ft2 = 1 yd2. The conversion for a cubic unit may be obtained by cubing (taking to the third power) the conversion for a unit of length. For instance since 3 ft = 1 yd, then 27 ft3 = 1 yd3. To convert square units in the metric system move the decimal twice as far as for a unit of length. Since 1 m = 100 cm, then 1 m2 = 10,000 cm2. The decimal is moved 4 places to the right instead of 2 places to the right. To convert cubic units in the metric system move the decimal three times as far as for a unit of length. Since 1 m = 100 cm, then 1 m3 = 1,000,000 cm3. The decimal is moved 6 places to the right instead of 2 places to the right. An alternative method of converting area and volume units is to just use the length conversions more than once as shown in the following examples: Example 8. Convert 10 in2 to cm2 given the conversion 1 in = 2.54 cm.

Answer: Multiply 2.54 cm 2.54 cm

10 1 in 1 in1 in 1 in

64.516 cm2


Example 9. Convert 600 in3 to ft3 using the conversion 1 ft = 12 in.

Answer: Multiply 1 ft 1 ft 1 ft

600 1 in 1 in 1 in12 in 12 in 12 in

0.347 ft3



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1. What would be an appropriate metric unit to measure the following?

a) the volume of a beverage can b) the length of a cat

c) the mass of an adult d) the length of a dictionary

e) the capacity of a soup can f) the length of an elephant

g) the mass of a truck

2. If a paper dollar has a mass of approximately one gram, is it possible to lift $1,000,000 in a) $1 bills? b) $10 bills? c) $100 bills?

3. If at a store meat costs $5.20 per kg, how much does 300 g cost?

4. Iron will melt at 5,432 0F. Find the equivalent Celsius temperature.

5. One nickel weighs about 5 g. How many nickels are in 1 kg of nickels?

6. Helium weighs about 0.0002 g per milliliter. How much would one liter of helium weigh?

7. Convert to Celsius

a) 15 0F b) 212 0F c) -40 0F d) 75 0F

8. Convert to Fahrenheit:

a) 10 0C b) 0 0C c) 100 0C d) -40 0C

9. Sue has a mass of 86 kg. How many grams is this?

10. Each day Libby takes a vitamin containing 18 mg of iron. How many grams of iron does she take in one year?

11. An analytical balance is used to measure the mass of four different tissue

masses. The masses are determined to be: 1.059 g, 0.984 g, 1.001 g, and 0.497 g.

a) What is the combined weight of the masses?

b) What is the average of the four tissue masses?


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12. To produce 1000 milliliters of a liquid suitable for culturing bacteria, 21 grams of culture medium are needed.

a) How many liters can be prepared from 975 grams of culture medium?

b) How many grams are required to prepare 5.750 L of solution?

13. If a certain spice costs $25 per kilogram, how much does 1 gram cost?

14. Convert

a) 20 miles to kilometers b) 1500 grams to ounces

c) 60 quarts to liters d) 4.5 cups to mL

e) 400 mL to teaspoons f) 600 cm to ft (convert to inches first)

g) 4000 m to miles (convert to kilometers first)

h) 60 lbs to kg

15. Still think the English system is much easier than the metric system? Try these! These are all English system units. Find them on the internet.

a) How many gills are in one peck?

b) How many yards are in one furlong?

c) How many US liquid gallons are in one barrel of crude oil?

d) How many US liquid gallons are in one barrel of beer?

e) How many US liquid gallons are in one barrel of whiskey?

f) How many grains are in one pound?

g) How many pounds of cotton in one bale?

h) How many herrings are in one warp of herrings?

i) How many yards of linen are in one bolt of linen?

j) How many pounds are in one stone?

k) How many mL are in one demijohn of champagne?


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1. a) cm3 b) cm c) kg d) cm

e) mL or L f) m g) tonne

2. Convert to lbs:

a) no b) not unless you are a weight lifter c) yes

3. $1.56

4. 3000 0C

5. 200

6. 0.2 g

7. a) -9.4 0C b) 100 0C c) -40 0C d) 23.9 0C

8. a) 50 0F b) 32 0F c) 212 0F d) -40 0F

9. 86,000 g

10. 6.57 g

11. a) 3.541 g b) 0.885 g

12. a) 46.4 L b) 120.75 g

13. $.025 or 2.5 cents

14. a) 32.2 km b) 53 oz c) 56.8 L d) 1064.25 mL e) 81.1 teaspoons f) 19.7 ft g) 2.48 mi h) 27.2 kg

Unit 4 Geometry and Units of Measurement Section 4.7 Unit review

367


Circle

2rA r is the radius

C d d is the diameter

Triangle

1

2A bh b is length of the base

h is the height

2 2 2a b c right triangle only c is the length of the hypotenuse

Parallelograms, rectangles, etc. A=b h b is the length of the base

h is the height

Trapezoid

1 2b b

A h2

1 2b and b are the lengths of the parallel sides

h is the height

Prisms and Cylinders V B H B is the area of the base figure

H is the height


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Solids that are pointed (cones, pyramids)

3

1V B H B is the area of the base figure

H is the height

Sphere

3

34V r r is the radius

Temperature Conversion

Step 1: Add 40 Step 2: Divide by 1.8 (F to C) or multiply by 1.8 (C to F) Step 3: Subtract 40

Unit Conversions

1 mi = 1.61 km 1 kg = 2.205 lbs 1 tonne = 1.102 tons 1 gal = 3.784 L 1 tsp = 4.93 mL 1 Tbsp = 15 mL 1 oz = 28.3 g 1 oz = 29.563 mL 4 qt = 1 gal 2 c = 1 pt 2 pt = 1 qt 16 fl oz = 1pt 1 mi = 5280 ft

Unit 4 Geometry and Units of Measurement Section 4.7 Unit review problems

369

1. Give both a metric and an English unit for each of the following:

Quantity English Unit Metric Unit a) mass or weight _________ _________ b) temperature _________ _________ c) length _________ _________ d) area _________ _________ e) volume _________ _________ f) capacity _________ _________

2. A paperclip may be used to illustrate three metric units. Fill in below:

Part of the paperclip Unit illustrated a) ________________ ________________ b) ________________ ________________ c) ________________ ________________

3. a) What is the freezing point of water in the metric system?

b) What is the boiling point of water in the metric system?

c) What is the mass of one cc of water?

d) What unit of volume is equal to one milliliter?

4. Identify each of the following angles as acute, obtuse, right or straight:

a) b)

c) d)


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5. a) What is an appropriate metric unit for the length of time it takes a

computer to perform a single operation (measured in billionths of a second)? b) What is an appropriate metric unit for the number of bytes of memory in a

computer (measured in billions of bytes)? c) What is an appropriate metric unit for the weight of a small can of beans?

d) What is an appropriate metric unit for the capacity of a container of orange juice?

e) What is an appropriate metric unit for the distance from Miami to New

York? f) What is an appropriate metric unit for the weight of a human being?

6. a) 0.076 m = ______cm

b) 44,000 mg = ______kg

c) 6500 dL = ______L

d) 0.084 km = ______mm

e) 100 0C = ______0F

7. A rectangular box measures 9 dm by 75 cm by 4 m.

a) Find its volume in cm3.

b) Find its capacity in L.

8. a) Draw a hexagon.

b) Draw a pentagon.

c) Draw a figure which is not a polygon.

d) Draw an octagon.

e) Draw a quadrilateral.


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9. For the triangle below,

a) Set up an equation to find x.

b) Solve the equation and find x with proper units.

10. The ends of the figure below are quarter circles. The top and the bottom are parallel to each other.

a) Find the area of the figure above and give proper units. Round your answer to the nearest whole unit.

b) Find the perimeter of the figure above and give proper units. Round your

answer to the nearest whole unit. 11.

In the figures above, the following angles are the same size:

A = F, B = D, C = E

a) Find x with proper units. b) Find y with proper units.


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12. A room has the floor plan shown below. All angles which look like right angles are right angles. Give units on answers.

a) Find the area of the room using proper units.

b) Find the volume of the room using proper units if the height of the ceiling is 9 ft.

c) Find the perimeter of the room using proper units rounded to the nearest

whole unit. 13. The left portion of the figure below is a half-circle, and the right portion is a

right triangle. Give correct units on answers.

a) Find the perimeter of the figure rounded to the nearest whole unit.

b) Find the area of the figure rounded to the nearest whole unit.

14. Find the volume of a pyramid with a 40 foot by 40 foot square base and a height of 30 feet, and give the answer using proper units.


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15. Find x.

16. Use similar triangles to find x rounded to the nearest whole unit.

17. a) Find the perimeter of the figure below, which is made up of a right triangle and two semi-circles, rounded to the nearest whole unit. b) Find the area of this figure rounded to the nearest whole unit.


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18. Find the area of the figure shown below.

19. a) Find the perimeter of the following figure.

b) Find the area of the following figure.

20. Name the following figures.


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21. You must pave a driveway which is 10 feet wide, 40 feet long and 6 inches

thick. Calculate how many cubic yards of concrete you must order if you can order only whole or half cubic yards?

22. Assume that the two horizontal lines below are parallel. Find the area of each

of the triangles shown.

23. Find the perimeter of the following figure in centimeters assuming that all angles are right angles.

24. Select the most appropriate metric unit for measuring a person’s height.

a) kilometer b) centimeter c) nanometer d) foot

25. Select the most appropriate metric unit for the weight of a flea egg.

a) kilogram b) gram c) milligram d) ounce


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26. Select the most appropriate metric unit for measuring the amount of paint in a

bucket.

a) kiloliter b) liter c) milliliter d) gallon

27. Select the metric unit that would probably be used at the gasoline pump.

a) gallon b) gram c) liter d) kiloliter

28. a) Find the area of the following figure if the cut out portion is one-fourth of a circle. Round your answer to the nearest whole unit, and give correct units with your answer.

b) Find the perimeter of the figure. Round your answer to the nearest whole

unit, and give correct units with your answer.

29. Use unit analysis to do the following conversions:

a) 3.4 miles = _______ft

b) 350 kg = _______lb

c) 800 g = _______oz

d) 50 in = _______cm

e) 850 km = _______mi

f) 512 oz = _______lb

g) 5.5 gal = _______qt

h) 54 ft = _______yd

i) 1300 mL = _______pt

j) 51 mi = _______km

k) 20 oz = _______g

l) 1024 fl oz = _______pt

m) 5 c = _______tbsp

n) 12 fl oz = _______tsp

o) 350 lb = _______kg


377

30. Parker has 50 cc of Xylocaine and two syringes on his back table. The doctor

needs 50% of this solution to inject in the nose and 25% of the solution to inject into the hip. How much of the Xylocaine should Parker draw up into each syringe?

31. A medicine is given according to the patient’s weight in kilograms. For every

kilogram, 1

2 cc of the medicine is needed. If the patient weighs 110 pounds,

how much of the medicine will be needed? 32. What is the measure of the angles below?

a.

b.


378

c.


379

1. Possible answers:

a) pound gram

b) 0F 0C

c) foot meter

d) square foot square meter

e) cubic foot cubic meter

f) quart liter

2. a) width cm

thickness mm

c) weight g

3. a) 0 0C b) 100 0C c) 1 g d) 1 cm3

4. a) acute b) straight c) right d) obtuse

5. a) nanosecond b) Gigabyte c) gram d) L or mL e) km f) kg

6. a) 7.6 b) 0.044 d) 84,000 e) 212 0F

7. a) 2,700,000 cm3 b) 2700 L


380

8. a) b)

c) d)

or

e)

9. a) 2x + 6x + 4x + 30 = 180 b) x = 12.50

10. a) 1,050,310 cm2 b) 5999 cm

11. a) 415 cm b) 346 cm

12. a) 2135 16 2 8 5 4 534 ft

2

b) (534)(9) = 4806 ft3

c) 12 + 10 + 2 + 8 + 2 + 17 + 16 + 30 + 2 24 5 = 103 ft

13. a)

2 2280

350 350 280 999.8229715 cm or 1000 cm2

b)

2

2 21401

280 210 60,187.60801 cm or 60,188 cm2 2


381

14. 16,000 ft3

15. 500

16. 43 m

17. a) 160 cm b) 1582 cm2

18. 7600 cm2

19. a) 90 cm b) 270 cm2

20. a) triangle b) rectangle c) pentagon d) hexagon e) parallelogram f) octagon

21. 31

7 yd2

22. All have an area of 150 cm2.

23. 1700 cm

24. b)

25. c)

26. b) or possibly c)

27. c)

28. a) 4468 cm2 b) 387 cm

29. a) 17,952

b) 772

c) 28.3

d) 127

e) 528

f) 32

g) 22

h) 18

i) 2.75

j) 82

k) 566

l) 64

m) 80

n) 72

o) 159

30. 25 cc’s and 12.5 cc’s

31. 110 lbs = 50 kg rounded to the nearest kg. Therefore, 25 cc’s of medicine will be needed.

32. a. 900 b. 1800 c. 1200


383

SUMMARY OF FORMULAS FOR MAT 155

THE FOLLOWING INFORMATION AND NO OTHER WILL BE FURNISHED ON THE FINAL EXAM.

Unit 2

Equation of a Line y=mx b where m is the slope of the

line, and b is determined by a point on the line. The point (0, b) is the y-intercept of the line.

Slope of a Line

2 1

2 1


run horizontal change x x

amount y changes when x increases by 1

Mean x

xn

x

N


384

Unit 3

I = PRT I = interest P = principal R = interest rate T = time

A = P + I A = amount including interest

Payment = A

N N = total number of payments

Compound interest

n×T

RA P 1

n

n = number of times per year money is compounded

yearly n = 1 semi-annually n = 2 quarterly n = 4 monthly n = 12 weekly n = 52 daily n = 365


2 N RAPR

N 1

N = total number of payments

APR = annual percentage rate

Amount of payment for an installment loan where the APR is given

-n×T

P amount borrowedAPRP×





385

Unit 4

Circle

2rA r is the radius

C d d is the diameter

Triangle

1

2A bh b is length of the base

h is the height

2 2 2a b c right triangle only c is the length of the hypotenuse

Parallelograms, rectangles, etc. A=b h b is the length of the base

h is the height

Trapezoid

1 2b b

A h2

1 2b and b are the lengths of the parallel sides

h is the height

Prisms and Cylinders V B H B is the area of the base figure

H is the height


386

Solids that are pointed (cones, pyramids)

3

1V B H B is the area of the base figure

H is the height

Sphere

3

34V r r is the radius

Temperature Conversion

Step 1: Add 40 Step 2: Divide by 1.8 (F to C) or multiply by 1.8 (C to F) Step 3: Subtract 40

Unit Conversions

1 mi = 1.61 km 1 kg = 2.205 lbs 1 tonne = 1.102 tons 1 gal = 3.784 L 1 tsp = 4.93 mL 1 Tbsp = 15 mL 1 oz = 28.3 g 1 oz = 29.563 mL 4 qt = 1 gal 2 c = 1 pt 2 pt = 1 qt 16 fl oz = 1pt 1 mi = 5280 ft

1.4 equations 47 1.7 unit review 85

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