410

45100

4891000

455610,000

45100

3 45100

1

3.1 Place Value and Word Names of Decimals

The word decimal is derived from the Latin word decem, which means “ten.” Our realnumber system is based on powers of 10. Thus, we refer to it as the decimal system ofnumeration. Decimals are numbers which may be expressed as a fraction with a denominatorthat is some power of 10. We commonly refer to the decimal part of a number as the partof the number to the right of the decimal point. An example is given below.

45.489 ���� � .489 is the decimal part.

Decimal parts are always equal to some proper fraction that has a denominator that is somepower of ten. Examples are given below.

0.4 is equal to .

0.45 is equal to .

0.489 is equal to .

0.4556 is equal to .

In general, any real number containing a decimal point is referred to as a decimal. Eachdecimal may be thought of as a mixed number. The part of the number to the left of thedecimal point is the whole number part, and the part to the right of the decimal point is equalto some fraction with a denominator that is a power of ten.

MATH FACTS ABOUT DECIMALS

���� Every decimal may be written as a fraction or mixed number.

���� The decimal point acts as the + sign between the whole number partand the decimal part.

���� Each decimal part may be written as a fraction with a denominatorequal to a power of ten.

Example: 3.45 = 3 + 0.45 = 3 + = .

099103

991000

12 991000

2

Writing Decimals as Fractions or Mixed Numbers

Any decimal may be written as a fraction or a mixed number where the fraction has adenominator that is a power of ten. The number of places to the right of the decimal pointindicates the power of ten in the denominator of the fraction. A technique for writing adecimal as a mixed number is summarized in the following procedure.

PROCEDURE TO CONVERT A DECIMAL INTO A FRACTION OR MIXED NUMBER

1. If the decimal contains a nonzero part to the left of the decimal point,rewrite the decimal as a whole number added to the decimal part.The decimal point acts as the + sign.

2. Write the decimal part over a power of ten as a fraction. The numberof places to the right of the decimal point will be equal to the powerof ten in the denominator of the fraction.

3. Remove the + sign to write the sum as a mixed number.

Example 1 Write 12.099 as a mixed number.

12.099 = 12 + 0.099 ���� 3 places

Since there are 3 places to the right of the decimal point, the fraction willhave a denominator of 10 or 1000.3

12 + 0.099 = 12 + = 12 +

This is equal to the mixed number .

Note that the left zero in the number 099 was dropped.

00870005100,000,000

870,005100,000,000

1 870,005100,000,000

04507100,000

4507100,000

78100

78100

70100

8100

710

8100

3

Example 2 Write 1.00870005 as a mixed number.

1.00870005 = 1 + 0.00870005 ���������� 8 places

Since there are 8 places to the right of the decimal point, the fraction willhave a denominator of 10 or 100,000,000 .8

1 + 0.00870005 = 1 + = 1 +

This is written as the mixed number, .

Example 3 Write the decimal 0.04507 as a fraction.

In this example, there is no whole number part. Thus, this decimal willconsist solely of a proper fraction. Since there are 5 places to the right of thedecimal point, the fraction will have a denominator of 10 = 100,000. 5

0.04507 = =

Place Value

The place values of decimal parts are tenths, hundredths, thousandths, and other fractionalparts with denominators that are powers of ten. For example, the place value of 7 in 0.78 istenths and the place value of 8 is hundredths. If 0.78 is written as a fraction, we obtain

0.78 = .

however, is equal to + = + .

In summary, a simple technique to identify the place value of a decimal digit consists ofcounting the number of places to the right of the decimal point that the digit is located. Thenumber of places is equal to the power of 10 in the denominator of the fraction thatrepresents this digit. This procedure is summarized on the following page.

1101

110

1105

1100,000

510

3100

91000

510

3100

91000

4

PROCEDURE TO IDENTIFY THE PLACE VALUE OF A DECIMAL DIGIT

1. Count the number of places to the right of the decimal point that the digit islocated.

2. The number of places is equal to the power of 10 in the denominator of thefraction that represents this digit.

Example 4 Find the place values of the digits 9 and 2 in 0.93712 .

The digit 9 is one place to the right of the decimal point. Thus, the fractionthat represents this digit is = .

The digit 9 is in the tenths place and the value of 9 is nine tenths.

The digit 2 is five places to the right of the decimal point. Thus, the fractionthat represents this digit is = .

The digit 2 is in the hundred-thousandths place and the value of 2 is twohundred-thousandths.

Example 5 Find the place values of each of the digits in 0.539 and write this decimal asa sum of three fractions.

The digit 5 is one place to the right of the decimal point. Thus, 5 is in thetenths place and the value of 5 is five tenths or .

The digit 3 is two places to the right of the decimal point. Thus, 3 is in thehundredths place and the value of 3 is three hundredths or .

The digit 9 is three places to the right of the decimal point. Thus, the 9 is inthe thousandths place and the value of 9 is nine thousandths or .

The decimal 0.539 may be written as + + .

67103

671000

671000

7000104

700010,000

3 4051000

5

Word Names of Decimals

The word name of a decimal may be obtained by writing the mixed number form of thedecimal and then writing the word name of the mixed number. This procedure is given here.

PROCEDURE TO WRITE THE WORD NAME OF A DECIMAL

1. Write the decimal as a mixed number.

2. Write the word name of the mixed number. Be careful to only usethe word “and” between the names of the whole number and thefraction.

Example 6 Write the word name of 45.067 .

Since there are 3 places to the right of the decimal point, the mixed numberis equal to

45 + = 45 + = 45 .

The word name of this mixed number is forty-five and sixty-seventhousandths.

Example 7 Write the write word name of 0.7000 .

There are 4 places to the right of the decimal point. Thus, this decimal isequal to = .

The word name is seven thousand ten-thousandths.

MATH FACT

���� When converting a decimal into a mixed number or a fraction, thenumber of zeroes in the denominator of the fraction will always beequal to the number of places to the right of the decimal point.

Example: 3.405 =

���������������� ���������������� 3 places = 3 zeroes

810

80100

800010,000

810

6

3.2 Listing Decimals in Order and Rounding

Adding or Deleting Zeros From the Far Right Side of a Decimal

If you write the decimals 0.8, 0.80, and 0.8000 as fractions you obtain

0.8 = 0.80 = 0.8000 = .

All of these fractions reduce to and thus 0.8 = 0.80 = 0.8000 . Because the addition of the extra zeroes at the far right side results in an equivalent fraction, the value of thedecimal is not changed. Likewise, deleting zeroes from the far right side of a decimaldoes not change the value of the decimal.

MATH FACT

���� Adding or deleting zeroes from the far right side of a decimal will notchange the value of the decimal.

Example: 0.09000 = 0.09 = 0.09000000

Note: Only zeroes on the far right can be added or deleted. In theexample given here, the zero to the left of 9 can not be deleted,nor can a zero be added to the left of 9.

Example 1 Write 0.07 as an equivalent decimal with three, four, five, and six places tothe right of the decimal point.

0.07 = 0.070 = 0.0700 = 0.07000 = 0.070000

Listing Decimals in Order of Value

When comparing fractions, it is necessary to write all of the fractions as equivalent fractionswith a common denominator. If we wish to list the decimals 0.42, 0.4102, 0.411, and 0.401in order from least in value to greatest in value, we also compare the fraction equivalents ofeach of the decimals. To make sure that the decimals have fraction equivalents with thesame denominators we first add zeroes to the far right of the decimals to make them the samelength and then compare the fraction equivalents of the decimals. This procedure is shownon the following page.

420010,000

410210,000

411010,000

401010,000

401010,000

410210,000

411010,000

420010,000

7

We add two zeroes to 0.42, one zero to 0.411, and one zero to 0.401 . Then we compare thefraction equivalents.

0.42 = 0.4200 =

0.4102 =

0.411 = 0.4110 =

0.401 = 0.4010 =

The list of fractions, in order from smallest in value to largest in value is

, , , .

Thus, the list of decimals, ordered from smallest to largest in value is

0.401, 0.4102, 0.411, 0.42 .

The procedure for comparing values of decimals is summarized here.

PROCEDURE FOR COMPARING DECIMALS

1. Add zeroes to the far right of the decimals so that they all have thesame number of places to the right of the decimal point.

2. Compare the fraction equivalents of the decimals.

Example 2 List the decimals 1.14, 1.1088, 1.149, and 1.1503 in order from smallest tolargest in value.

Rewrite all of the decimals so that they have 4 decimal places. Add zeroesto the decimals and then write each as its equivalent mixed number. This isdone on the following page.

1 140010,000

1 108810,000

1 149010,000

1 150310,000

8

1.14 = 1.1400 = (second smallest)

1.1088 = (smallest)

1.149 = 1.1490 = (third smallest)

1.1503 = (largest)

Comparing these mixed numbers, we see that the smallest decimal is 1.1088,the second smallest is 1.14, the third smallest is 1.149, and the largest is1.1503 . The list from smallest to largest is 1.088, 1.14, 1.149, 1.1503 .

Rounding Decimals

To round a decimal, use nearly the same procedure that was used to round whole numbersin Chapter One. This procedure is given here.

PROCEDURE TO ROUND A NUMBER

1. Locate the digit to be rounded.

2. Look at the digit to the right. If the digit to the right is greater than orequal to 5, then add 1 to the digit that is rounded. If the digit to theright is less than or equal to 4, then leave the first digit as it is.

3. When rounding decimal places to the right of the decimal point, donot include any digits to the right of the rounded digit.

Example 3 Round 3.40522 to the nearest hundredth.

Add 1 to 0 �3.40522 rounds to 3.41

� Look at 5

The digit in the hundredths place is 0. Look at the digit to the right which is5. Since 5 is greater than or equal to 5, we add 1 to 0 and round up to 3.41,and we do not include any digits to the right of the rounded digit.

9

Example 4 Round 2.999804 to the nearest thousandth.

Add 1 to 9 �2.999804 rounds to 3.000 � Look at 8 The digit in the thousandths place is the third 9 to the right of the decimalpoint. Since the next digit 8 is greater than or equal to 5, we add 1 to 9.Adding 1 to 9 results in 2.999 rounding up to 3.000 . In effect, the followingaddition is performed.

2.999 � Note that the digits beyond the rounded digit are dropped.+.001�����3.000

Example 5 Round 104.032 to the nearest tenth and also round to the nearest hundred.

To round to the nearest tenth, we look at the digit to the right of 0 in thetenths place. Since 3 is less than 5, nothing is added to 0 and the digits to theright of zero are dropped.

Leave 0 as it is �104.032 rounded to the nearest tenth is 104.0 � Look at 3

To round to the nearest hundred, we look at the digit to the right of 1 in thehundreds place. Since 0 is less than 5, nothing is added to 1 and zeroplaceholders are inserted for the ones and the tens places. No decimal partis included.

Leave 1 as it is�

104.032 rounded to the nearest hundred is 100. � Look at 0

MATH FACT

���� When rounding to a whole number place, never include a decimal point orany decimal places.

10

Rounding and Measurement

When the decimal 12.3004 is rounded to the nearest thousandth, the result is 12.300 . Onemight say that we could write this result as 12.3, but this would be incorrect when 12.3004represents a measurement. The rounded result of 12.300 implies that the measurement isaccurate to a thousandth, however 12.3 implies that the measurement is accurate only to thenearest tenth. For example, if a chemistry student obtained the mass of a compound on anelectronic balance as 12.3004 grams and rounded this result to a thousandth of a gram, theywould report this mass as 12.300 grams. On the other hand, if a contractor used 12.25 cubicyards of cement on a job and rounded this result to the nearest tenth of a cubic yard, theywould report this amount as 12.3 cubic yards. These two amounts (12.300 and 12.3) areequivalent, yet the extra zeroes in 12.300 imply a greater degree of precision.

Example 6 A fuel oil truck delivers fuel oil to you through a hose running from the backof the truck. In a heating season your fuel oil tank is filled three times: oncewith 180.8 gallons of oil, once with 121.32 gallons of oil, and once withabout 85 gallons of oil. If you were calculating the total amount of heating oilused, would you report 387.12 gallons, 387.120 gallons, 387.1 gallons, or 387gallons?

Since these measurements, especially the measurement of 85 gallons, are notvery precise, it would be appropriate to report this sum as 387 gallons.

Example 7 In a chemistry experiment, you weigh out 3.7000 grams of a substance on astate-of-the-art balance and then divide this amount into two equal amountsof 1.8500 grams. Should you report this as 1.85 grams, 1.9 grams, or 1.8500grams?

Since the measurement device in this example is very precise, you wouldwant to report this mass as 1.8500 grams.

11

3.3 Operations with Decimals

Addition and Subtraction of Decimals

The process of adding and subtracting decimals consists of placing the decimals over eachother so that the decimal points line up, adding zero placeholders at the far right so that allof the decimals have the same number of decimal places, and then adding or subtracting fromright to left as if the decimals were whole numbers. This procedure is summarized here.

PROCEDURE FOR ADDING OR SUBTRACTING DECIMALS

1. Place the decimals in column form so that the decimal points line up.

2. Add zeroes to the far right so that all of the decimals have the samenumber of places to the right of the decimal point.

3. Add or subtract from right to left as if the decimals were wholenumbers. Insert the decimal point in the answer in the same locationas in the decimals added or subtracted.

Example 1 Add 2.3 + 4.909 + 1.01

Place these decimals in column form so that the decimal points line up.Then, add zeroes to the far right of the decimal points to make all of thedecimal parts the same length.

2.3 2.300 4.909 = 4.909+ 1.01 + 1.010�������� ��������

Now, add these numbers in the same way you would add 2300+4009+1010. 1

2.300 Note that when the placeholders 3 and 9 are added, the 4.909 result is 12 tenths. 2 is written in the tenths placeholder+ 1.010 location and 1 (ten tenths) is carried to the ones column.�������� 8.219

12

Example 2 Subtract 3.6 � 2.556 .

Place these decimals in column form so that the decimal points line up.Then, add zeroes to the far right of 3.6 so that both decimals have the samenumber of places.

3.6 3.600� 2.556 � 2.556�������� ��������

Now subtract these numbers in the same way that you would subtract thewhole numbers 3600 � 2556 .

9 5 1010

3.600 Note that the same borrowing procedure is used as in the� 2.556 subtraction of 3600 � 2556.�������� 1.044

Example 3 Add 3 + 2.08 + 6.1 .

In this example, 3 is not a decimal but a whole number. 3 can be written asthe decimal 3.00 . Also, 6.1 can be written as 6.10 .

3.00 The addition is performed in the same way as the 2.08 addition 300 + 208 + 610.+ 6.10������� 11.18

MATH FACT

���� A whole number may always be written as an equivalent decimal byadding a decimal point to the right of the ones place and adding asmany zeroes as desired to the right of the decimal point.

Example: 25 = 25.0 = 25.0000

1 2100

3 510

102100

× 3510

35701000

1 2100

3 510

35701000

35701000

3 5701000

13

Multiplying Decimals

When we multiply × we convert the mixed numbers into improper fractions and

multiply to obtain the result

= .

Since = 1.02 and = 3.5, we could say that 1.02 × 3.5 = . But is

equal to the mixed number which in decimal form is 3.570 . Thus, 1.02 × 3.5 = 3.570.

In this example, the decimals multiplied contained 2 decimal places and 1 decimal place.The answer contained 2 + 1 or 3 decimal places. Likewise, in the fraction multiplication, thefractions contained a denominator that was ten to a power of 2 and ten to a power of 1. Thefraction answer contained a denominator of ten to a power of (2 + 1) = 3. The final answerof 3.570 could have been obtained by multiplying 1.02 × 3.5 as if they were whole numbersand placing the decimal point so that the answer contained 3 places. A general procedure fordecimal multiplication is given here.

PROCEDURE FOR DECIMAL MULTIPLICATION

1. Multiply the decimal numbers as if they were whole numbers.

2. Count the number of decimal places in each factor, and add thesenumbers to find the sum of the decimal places. The answer will havea number of decimal places equal to the sum.

Example 4 Multiply 3.05 × 2.012

Multiply 3.05 × 2.012 as you would multiply 305 × 2012. This result is613660. We must insert the decimal point in this answer so that there are 5decimal places.

3.05 × 2.012 = 6.13660 2.012 ��� ���� ������ × 3.052 places + 3 places = 5 places �������

10060 603600 �������� 6.13660

14

Example 5 Multiply 5 × 4.2 × 3 .

This multiplication is performed in the same manner as the multiplication 5× 42 × 3. The total number of decimal places in the answer is only 1 sincethe whole numbers 5 and 3 have no decimal places.

5 × 4.2 × 3 63.0� � � �0 places + 1 place + 0 places = 1 place in answer

MATH HINT

���� When multiplying decimals, first estimate the answer if possible.Estimate by rounding the decimals to whole numbers and thenmultiplying. Compare your estimate to the actual answer.

Example 6 Estimate 2.02 × 5.01. Then multiply the numbers out to find the actualanswer.

2.02 rounds to 2 and 5.01 rounds to 5. Thus, the estimate is 2×5 = 10.

The actual answer has 4 decimal places and is 10.1202 .

5.01 If you were to obtain an answer significantly different

× 2.02 from the estimate, you would know that an error � � � � occurred. 1002

100200 � � � � � 10.1202

Multiplying and Dividing by Powers of Ten

When a decimal is multiplied by a power of 10, the decimal point is moved to the right bythe number of places indicated by the power on 10. Also, when a decimal is divided by apower of 10, the decimal point is moved to the left by the number of places indicated by thepower on 10. These procedures were outlined in Chapter One and are given again here.

MULTIPLICATION BY A POWER OF TEN

���� When multiplying by a power of ten, move the decimal point to theright by the number of places indicated by the exponent. Note thatfor whole numbers the decimal point must be inserted to the right ofthe ones place.

15

DIVISION BY A POWER OF TEN

���� When dividing by a power of ten, move the decimal point to the leftby the number of places indicated by the exponent. Note that forwhole numbers, the decimal point must be inserted to the right of theones place.

Example: 235 ÷ 10 = 235.0 ÷ 10 = 0.00002357 7

Example 7 Divide 3.455 ÷ 10 .6

3.455 ÷ 10 = 0.0000034556

��������� 6 places

Example 8 Multiply 456.003 × 10 .7

456.003 × 10 = 4560030000 = 4,560,030,0007

���������� 7 places

Division by a power of 10 is mathematically equivalent to multiplication by the negativeof that power of 10. For example, 4.52 ÷ 10 = 4.52 × 10 . 5 �5

MATH FACT

���� Division by a power of 10 is equal to multiplication by 10 to thenegative of that power.

Example: 8.93 ÷ 10 = 8.93 × 106 ����6

Example 9 Multiply 5.67 × 10 .�4

5.67 × 10 = 5.67 ÷ 10 = 0.000567�4 4

16

Scientific Notation

Scientific notation is a method of writing very large and very small numbers by using powersof 10. For example, the number 430,000,000,000 may be written as 4.3 × 10 . Using11

scientific notation provides a way to write extremely large and small numbers in a compactform that can be input into a scientific calculator. The procedure for writing a number inproper scientific notation is given here.

PROCEDURE TO WRITE A NUMBER IN SCIENTIFIC NOTATION

1. Move the decimal point so that there is exactly one non-zero digit tothe left of the decimal point.

2. Count how many places you moved the decimal point. This numberis the power you place on 10.

3. Multiply by 10 to a positive power when representing numbers largein value. Multiply by 10 to a negative power when representingnumbers small in value.

Example 10 Represent 545,000,000,000,000 by using scientific notation.

First, move the decimal point so that there is 1 non-zero digit to the left of thedecimal point. Count how many places the decimal point is moved.

545,000,000,000,000 = 5.45000000000000 × ? ������������������

14 places

Since the number of places = 14, the power on 10 is 14. Also, since this isa number large in value, we multiply by a positive power of ten.

545,000,000,000,000 = 5.45 × 1014

Note that the zeroes at the far right of 5.45 were deleted.

17

Example 11 Represent the number 0.000000002356 with scientific notation.

Move the decimal point so that there is 1 non-zero digit to the left of thedecimal point. Count how many places the decimal point is moved.

0.000000002356 = 000000002.356 × ? ������������ 9 places

Since the number of places = 9, the power on 10 is 9. Also, since this is anumber small in value, we multiply by a negative power of ten.

0.000000002356 = 2.356 × 10�9

Note that the zeroes to the left of 2 were deleted.

Scientific Calculators and Scientific Notation

In order to make good use of scientific notation, it is essential to know how to input thesenumbers into a scientific calculator and interpret the results. The following guidelines applyto most scientific calculators.

SCIENTIFIC NOTATION AND YOUR CALCULATOR

���� A number such as 2.4 × 10 will be displayed as 2.4 E 03 or 2.4 . 3 0 3

���� A number such as 7.9 × 10 will be displayed as 7.9 E����08 or 7.9 .����8 ����0 8

���� Input 5.6 × 10 on a scientific calculator as 5 . 6 Exp 1 1 .11

���� Input 7.9 × 10 on a scientific calculator as 7 . 9 Exp 4 +/- .����4

Example 12 If you multiplied 50,000 × 70,000,000 on a scientific calculator, how wouldthe answer of 3.5 × 10 be represented?12

This would be shown as 3.5 or 3.5 E12 . 12

1.375 6.85

51815

35350

25

1.2563 3.770

3761715

20182

18

Dividing Decimals

When a decimal is divided by a whole number using the long division process, the divisionis performed as if the divisor and the dividend were whole numbers. The decimal point inthe answer is located directly above the decimal point of the dividend.

In this example, the decimal point in 1.37 is directly above the decimal pointin 6.85 . The answer to this division could be estimated by performing thedivision 7 ÷ 5 = 1 or 1 r 2. From this estimation, we can see why thedecimal point must be placed after 1 in the answer. The procedure fordividing a decimal by a whole number is given here.

PROCEDURE FOR DIVIDING A DECIMAL BY A WHOLE NUMBER

1. Perform the division as if the divisor and the dividend were wholenumbers. Add zeroes to the far right of the decimal dividend ifneeded.

2. The decimal point in the answer is located directly above the decimalpoint of the dividend.

3. Continue the long division process so that there is one decimal placemore than what is desired in the rounded answer. This extra placewill be used for rounding.

4. Round off the answer to the desired decimal place.

Example 13 Perform the division 3.77 ÷ 3 and round the answer to the nearest hundredth.

Note that a zero placeholder was added to 3.77 and it waswritten as 3.770 . This division was performed in the sameway as the division 3770 ÷ 3 = 1256. The decimal point inthe answer was placed directly above the decimal point in3.770.

Rounding 1.256 to the nearest hundredth results in the final answer of 1.26.

0.02254 0.0900

810820200

0.320.7 0.32 × 10

0.7 × 103.27

.4577 3.200

28403550491

19

Example 14 Perform the division 0.09 ÷ 4 and round the result to the nearest thousandth.

Two zero placeholders are added to 0.09 and it is written as0.0900 . This division is performed in the same way as thedivision 900 ÷ 4 = 225. The decimal point in the answer isplaced directly above the decimal point in 0.0900 .

Rounding 0.0225 to the nearest thousandth results in the answer 0.023 .

Example 15 Perform the division 0.32 ÷ 0.7 and round the result to the nearest hundredth.

In this example, we are not dividing by a whole number. We can, however,change the form of this problem by moving the decimal point one place to theright in both the divisor and the dividend.

Since 0.32 ÷ 0.7 = , we may multiply the numerator and the

denominator of this fraction by 10 to obtain = which is 3.2 ÷7.

Thus, we perform the equivalent division 3.2 ÷7 .

Two zero placeholders are added to 3.2 and it is written as3.200 . This division is performed in the same way as thedivision 3200 ÷ 7 . The decimal point in the answer is placeddirectly above the decimal point in 3.200 .

Rounding 0.457 to the nearest hundredth results in 0.46 .

In Example 15, the divisor was not a whole number. The division was rewritten as anequivalent division by moving the decimal points the same number of places in both thedivisor and the dividend. This technique is used in a general procedure for dividingdecimals. The procedure is given on the following page.

2.9 0.03 29 0.3

.010329 0.300

29100

1008713

0.4 4.1 4 41

10.254 41.00

41010820200

20

PROCEDURE FOR DIVIDING ANY TWO NUMBERS

1. If the dividend is a whole number, insert a decimal point and as manyzeroes after the decimal point as are needed.

2. If the divisor is not a whole number, make it into a whole number bymoving the decimal point to the right in both the divisor and thedividend the same number of places.

3. Use the procedure for dividing a decimal by a whole number.

Example 16 Perform the division 0.03 ÷ 2.9 and round the answer to the nearest

thousandth.

First move the decimal points one place to the right in both the divisor andthe dividend.

=

Now, use the procedure for dividing a decimal by a whole number.

The final answer, rounded to the thousandths place, is 0.010 .

Example 17 Perform the division 4.1 ÷ 0.4 and round the result to the tenths place.

First move the decimal points one place to the right in both the divisor andthe dividend.

=

The rounded answer is 10.3 .

21

MATH FACT

���� Before dividing decimals, estimate your answer, if possible, byrounding decimals to whole numbers. If the estimate is not close tothe actual answer, there is probably an error in the actual calculation.

Example 18 If we divide 89.397 ÷ 9.9 and obtain a result of 0.0903, how would anestimate of this answer indicate that an error was made?

The estimate of this division is 90 ÷ 10 = 9. The correct result of dividing89.397 ÷ 9.9 is 9.03 .

Order of Operations

Order of operations for decimals is the same as it is for whole numbers. The order ofoperations is given again here.

ORDER OF OPERATIONS

1. Perform all operations within parentheses first.

2. Perform exponent operations before multiplying, dividing, adding orsubtracting.

3. Divide and multiply from left to right in the expression.

4. Subtract and add from left to right in the expression.

Example 19 Evaluate 4.2 + 3.1 ÷ 0.1 .2

First, multiply 0.1 by itself.

4.2 + 3.1 ÷ 0.1 = 4.2 + 3.1 ÷ (0.1 × 0.1)2

= 4.2 + 3.1 ÷ 0.01

= 4.2 + 310

= 314.2

(3.2 � 1.12) � 3.4 � 2.0 × 2.10.1 × 2.0

(3.2 � 1.12) � 3.4 � 2.0 × 2.1 ÷ 0.1 × 2.0

(3.2 � 1.12) � 3.4 � 2.0 × 2.1 ÷ 0.1 × 2.0

4.41 � 3.4 � 2.0 × 2.1 ÷ 0.1 × 2.0

4.41 � 3.4 � 2.0 × 2.1 ÷ 0.1 × 2.0

4.41 � 3.4 � 4.2 ÷ 0.2

1.01 � 4.2 ÷ 0.2

5.21 ÷ 0.2

22

Example 20 Evaluate

Note that the fraction bar implies division. This expression is equivalent to

.

First square 1.1 within the parentheses , then add this result to 3.2 . Note that1.1 = 1.1 × 1.1 = 1.21 and 1.21 + 3.2 = 4.41 .2

=

Next, perform the multiplication operations. Then, add and subtract from leftto right. Finally, divide the two results.

=

=

=

= 5.21 ÷ 0.2

= 26.05

37

37

.42857 3.0000

282014

605640355

4 313

4 313

313

313

.230713 3.0000

264039100

100919

23

3.4 Converting Fractions into Decimals

Since a fraction bar may always be interpreted as a division sign, the procedure forconverting a fraction into a decimal consists of dividing the numerator by the denominatorand then rounding the result. This procedure is given here.

PROCEDURE TO CHANGE A FRACTION INTO A DECIMAL

1. Divide the numerator by the denominator.

2. Carry out the division one place beyond the digit that is to berounded.

3. Round the result.

Example 1 Write the fraction as a decimal rounded to the thousandths place.

= 3 ÷ 7

The rounded result is 0.429

Example 2 Write the mixed number as a decimal rounded to the thousandths place.

Since = 4 + , convert the fraction into a decimal rounded to the

thousandths, and add this decimal to 4.

The rounded decimal is 0.231 . 4 + 0.231 = 4.231 .

370

�5

11

.0428570 3.00000

280200140

60056040035050

.454511 5.0000

4460555044

605550

15

�12

�1

10

15

�12

�1

102

10�

510

�1

108

10

0.25 1.0

100

0.52 1.0

100

0.110 1.0

100

810

37

����7

11����

1113

24

Example 3 Add by converting each fraction into a decimal rounded to thethousandths place, and then add the decimals.The two divisions are

The rounded results add to 0.043 + 0.455 = 0.498

Example 4 Add and obtain a fraction result. Then add by converting eachfraction into a decimal, and then add the decimals. Compare your results.

= =

The decimal equivalents of the fractions are

These decimals add up to 0.2 + 0.5 + 0.1 = 0.8 .

These results are equal since 0.8 = .

MATH FACT

���� Fractions may be added or subtracted on a scientific calculator inone step by using the +, ����, and ÷ keys.

Example: is input as 3 ÷ 7 + 7 ÷ 11 ���� 1 ÷ 113 = .

The result shown on the calculator is 1.0560855074. . .

445511,231

�61

117

29,000879,003

�3441

13,417

181

4

18

14

181

41

32

132

132

132

25

Example 5 Use a scientific calculator to add and round the answer to thenearest thousandth.

This is entered as 4455 ÷ 11231 + 61 ÷ 117 = and the displayed result is.918037452807 which rounds to 0.918 .

Example 6 Use a scientific calculator to add and round the answer tothe nearest thousandth.

This is entered as 29000 ÷ 879003 + 3441 ÷ 13417 = and the displayed resultis .289457601676 which rounds to 0.289 .

MATH REMINDER

���� A scientific calculator uses the correct order of operations. Mostnon-scientific calculators do not. A scientific calculator willperform the divisions in 1 ÷ 2 + 2 ÷ 5 before adding and return acorrect result of 0.9.

���� A non-scientific calculator will usually perform all operations fromleft to right and will incorrectly calculate 1 ÷ 2 + 2 ÷ 5 as 0.5 .

Example 7 A gas and oil mixture for a lawn mower requires of a gallon of oil for each

gallon of gas. If of one gallon of gas is added to the mower, how many

ounces of oil must be added? Note that 1 gallon is equivalent to 128 ounces.

Since of a gallon of oil is added to 1 gallon of gas, × gallons of oil are

added to of a gallon of gas. This means that of a gallon of oil is added.

of one gallon is equivalent to of 128 ounces.

is converted into the decimal 0.03125 .

0.03125 × 128 ounces = 4 ounces of oil