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Chapter 1 Technical Mathematics for Today's World Page 7

Chapter 1

Basic Skills

Application Preview

Solar energy is an ever more important component in our energy needs. In order for solar panels to

be most effective they should be angled so that the panel's face is perpendicular to the sun. See

figure 1. Unfortunately the only way this can be accomplished is for the panel to both rotate

throughout the day (follow the sun) and change its inclination (angle off the horizontal) throughout

the year to accommodate the tilt of the Earth as it orbits around the sun. A simple/cheap alternative

is to fix the panel at one 'average' angle.

Aligning

Solar

Panels

Sun

figure 1

N

S

Sun23.5°

44.0°

Equator

Position of Sun

and Earth

on Winter Solstice

44° latitude

?

figure 2

What mathematical skills are needed to solve such a problem? Determining the optimum angle for

solar panels is just one of the skills you should attain upon completion of chapter 1.

Overview

Basic Skills is intended to be a review of material already studied in a prior course (e.g. pre-algebra) that

will be useful in this course as well as the natural extension to some concepts not necessarily covered in

a pre-algebra course. The chapter covers basic arithmetic skills beginning with a review of common

mathematical terminology and notation. Classification of numbers, order of operations, computation

with fractions, powers of 10, tape measures and rulers, rounding, some geometry and graphing basics

and working with formulas is then discussed. The chapter concludes with a discussion of using the TI-

83 calculator.

Technical Mathematics for Today's World Chapter 1

Chapter 1 Topics

1.0 Are You Ready? ............................................................................................... 8

1.1 Notation & Vocabulary .................................................................................. 11

1.2 Rules for Real Number Arithmetic ................................................................ 18

1.3 Working with Fractions ................................................................................. 28

1.4 Some Calculator Tips (TI-83 Family) ............................................................ 36

1.5 Squares, Cubes & Roots and Exponential Notation....................................... 43

1.6 Powers of 10 and Scientific Notation............................................................. 46

1.7 Absolute Value ............................................................................................... 54

1.8 Reading a Ruler or Tape Measure .................................................................. 56

1.9 Rounding, Accuracy and Significant Digits .................................................. 59

1.10 Formulas and Substitution ........................................................................... 72

1.11 Some Geometry Basics ................................................................................ 78

1.12 Some Graphing Basics ................................................................................. 84

Chapter 1 Test ....................................................................................................... 96

Chapter 1 Project ................................................................................................... 99

1.0 Are You Ready?

The following Are You Ready? quiz is designed to briefly test your understanding of basic mathematical

notation & vocabulary, computation with fractions, decimals and whole numbers, substitution, percents,

basic geometry, using ratios and proportions and a few other odds and ends. It is essentially a

prerequisite quiz for the text and you should use the quiz results to determine the amount of review

and/or additional help you'll need now, at the beginning of this text.

The quiz is not necessarily a timed quiz and can easily be self-corrected. Obviously if you are receiving

outside help while trying the quiz, looking up information you are not sure about and giving your self

unlimited time you should be evaluating yourself accordingly. The purpose of this quiz is solely to let

you get realistic feedback regarding how much help you need now before crunch time comes.

We suggest you try the quiz, self correct it and then focus your review on the material that you found

difficult or confusing or for which you got an incorrect result. If the text review is insufficient for a full

understanding see your instructor for additional help. Be careful not to conclude that you understand the

material just by scanning the answer key.


Are You Ready?

Try the entire quiz first without seeking help. Self correct the quiz and then seek help. Unless otherwise

indicated, write your answers as an exact integer or as a simplified fraction.

1) (a) T F (3 + 4)·5 = [3 + 4]·5 (b) T F 4

2 = 44

(c) T F 1

3 = 0.33 (d) T F 2 + 3/5 =

2 + 3

5

(e) T F 3.609 rounds to 3.7 (f) T F 6 sq-ft = 6'2

(g) T F 5 − 5

3 − 3 = 0 (h) T F

0

4,768 = 0

2) 2,345.6789 has a 7 in the place and a 3 in the place.

3) (a) Write numerically, ninety three million. Include commas.

(b) Write algebraically, the square of the sum of A and B.

4) Reduce the fraction: (a) 12

66 = (b)

14

112 =

5) Convert to a reduced fraction: (a) 0.25 = (b) 0.4 =

(c) 6.12 = (d) 0.33 =

6) Convert to a mixed number: (a) 73

3 = (b)

933

5 =

Convert to an improper fraction: (c) 7 3

5 = (d) 4 2

3 =

7) Compute and give your answer as a fraction:

(a) 2

3 +

5

3 = (b)

7

15 +

7

12 = (c) 7

3

5 – 2 1

3 =

8) Compute and give your answer as a fraction:

(a) 2

3 ·

5

8 = (b)

2

3 ÷

5

12 = (c) 1

3

5 · 2 1

3 =

9) Simplify the calculation by hand to a single value:

(a) (3 + 4)10 – 5 = (b) 8·32 + 7 = (c) 10 ÷ 12 ÷

5

6 =

In 10 – 12, use a calculator to obtain a single value accurate to 2 decimal places.

10) (a) 5476 = (b) 8 · 32

7 + 4 = (c)

16.5 · 12.2

14.8 · 7.5 =

11) Use substitution to evaluate V: P = 2.6, Q = 1.5; V = 3P + 4Q. V =

12) Use substitution to evaluate T: D = 9.8, P = 0.15; T = D

1 + P2 . T =

13) Convert to a percent: (a) 0.4 = (b) 1.25 = (c) 0.0025 =

14) Convert to a decimal: (a) 12% = (b) 1 1

4 % = (c) 150% =


15) (a) 20% of 640 is . (b) 34 correct out of 40 is %.

(c) For a $482.00 gross pay what is the net pay after a 22% tax deduction?

16) Use the symbols R for the radius, D for the diameter and C for the

circumference to label this circle diagram:

Give the formulas for the circumference (when the diameter is known) and the

area (when the radius is known).

Formula: C = Formula: A =

17) (a) Find the circumference of a pipe with an 8" diameter accurate to 2 decimal places. C =

(b) Find the area (nearest sq-ft) for a traffic circle with a 120 ft radius. A =

18) Give the formula for the area of a

triangle when the base, B and height, H

are known and then compute the area of

these triangles.

a)

12

18

b)

62

883934

Formula: A = (a) Area = (b) Area =

19) Compute x for each ratio/proportion problem:

(a) 3

8 =

x

24 (b)

3

8 =

x

12 (c)

3

x =

5

12

20) (a) Compute the average fuel consumption: 356 miles driven using 14.2 gallons of gas.

(b) Compute the average grade for these three scores: 79, 82, 93.

21) (a) Convert to decimal form: 60% (b) 125% (c) ½%

1.0 Answers-Are You Ready Chapter 1

1) T, F, F, F, F, F, F, T

2) hundredths, hundreds

3) 93,000,000, (A + B)2

4) 2/11, 1/8

5) 1/4, 2/5, 6 3/25, 33/100

6) 24 1/3, 186 3/5, 38/5, 14/3

7) 7/3, 21/20, 79/15

8) 5/12, 8/5, 56/15

9) 65, 79, 1

10) 74, 6.55, 1.81

11) 13.80

12) 9.69

13) 40%, 125%, 0.25%

14) 0.12, 0.0125, 1.50

15) 128, 85%, $375.96

16) **

17) 25.13", 45,239 ft2

18) A=(½)B·H, 108, 1,054

19) x=9, x=9/2, x=36/5=7.2

20) 25.07mpg, 84.67

21) 0.60, 1.25, 0.005

16)

DR

C

C = πD A = πR

2


1.1 Notation & Vocabulary

Why Study Technical Mathematics? Practically speaking, mathematics is a non-ambiguous way to

express, define and explore questions in the real world. For mathematics to be effective, we need

precise terminology and notation and explicitly defined rules for dealing with that notation.

Furthermore, all rules must be consistent with each other and any new rules must be consistent with all

previous rules.

Unfortunately, there are ambiguities throughout mathematics though many can be resolved from their

context. For instance, some might write 2 x 3 while another might write 2 · 3 for the same calculation.

It is important to become accustomed with the common variations of popular terminology and notation

and adopt a style which is consistent and practical. Thus, we begin our discussion of Technical

Mathematics by examining some common symbols and vocabulary.

1.1.1 Common Symbols and Notation

Here are some common mathematical symbols and notation. You should have seen most of these before

now. We will use many of these in this course though not necessarily in this first Chapter. However, a

couple of these are commonly misused. Can you spot them? Glance through them and circle the ones

you are not sure about. Then ask your instructor for further clarification.

Parentheses

( ) rounded brackets Used to control order of operations, arithmetic grouping and

arithmetic on the TI-83. Also used for ordered pairs (graphing).

[ ] square brackets

When written, they can be used in the same manner as rounded

brackets. They CANNOT be used for arithmetic on the TI-83.

Also used to denote intervals. e.g. [min-value, max-value]

{ }

curly brackets

braces

set brackets

When written, they can be used in the same manner as rounded

brackets. They CANNOT be used for arithmetic on the TI-83.

Also used to denote sets. e.g. {0, 1, 2, 3, …} denotes the set of

whole numbers.

a{b[(c) + (d)] + e}

[a(b + c)

d(e + f)]

2

nested parentheses

Mixing the style of parentheses is convenient to visually align

nesting parentheses in complex expressions.

(a)(b), a(b), (a)b multiplication Parentheses can be used to indicate multiplication

(x, y), (x, y, z) ordered pairs Only rounded brackets are used for Cartesian coordinates

(-a) + (-b) brackets as

separators Parentheses are often used to separate negative numbers

Arithmetic

sum of a and b

a + b addition These are all equivalent forms of the same expression

difference of a and b

a minus b

a − b

subtraction These are all equivalent forms of the same expression


quotient of a and b

a/b = a

b = a÷b

division These are all equivalent forms of the same expression

product of a and b

a*b, a·b, a×b, ab

(a)(b), a(b), (a)b

multiplication These are all equivalent forms of the same expression.

3 4 = 3×4

a space can

indicate

multiplication

Though technically valid, it is not a good idea to use a space to

indicate multiplication. Why?

23, 2^3 exponentiation

The caret is used in many calculators and computer programs to

indicate exponentiation.

a , a + b square roots The end of the square root symbol indicates the extent of the square

root operation. e.g. 9 + 16 = 3 + 16 = 19, 9 + 16 = 25 = 5

38 cubed root 3

8 = 2 because 23 = 8.

1½ = 3/2 = 3

2 = 1.5 fractions These are all equivalent forms of the same number

inequality Used to show two expressions are not equal.

DNE

empty set

does not exist

Used to indicate no solution, or an invalid calculation such as

division by zero. Do NOT write the number zero as

≈ approximate Used for inexact results. e.g. π ≈ 3.14

>, ≥

<, inequality

Greater than, Greater than or equal. e.g. 7 > 6, 4 ≥ 4

Less than, Less than or equal. e.g. 9 ≤ 12, -3 ≤ -3

0 < x 10 between x is between 0 and 10. However, x is never equal to zero but could

be equal to 10.

=? are these equal Useful notation when checking the solution to an equation.

% percent Percent means divide by 100. e.g. 40% = 40/100

a ± b multiple numbers A convenient way to indicate two numbers. 4 ± ½ is the same as

the pair 3 ½, 4 ½ .

Measurement

6 ft 4 in, 6' 4", 6'-4" feet - inches A dash is not always a subtraction. It's common to use a dash in

measurements. e.g. sq-ft is NOT indicating subtraction.

6 sq-ft , 6 ft2, 6 square feet

6'2 is NOT a valid notation for six square feet.

1 ft 4 in2 is NOT a valid notation for a 1' 4" x 1' 4" square.

a ± b range of values A convenient way to indicate a range of values. 2.0 ± 0.1 is the

range 1.9 to 2.1.

5#

#5

five pounds

the number five

# (the pound sign) is used for pound units, for listing numbers or as

an abbreviation for 'number'.

° ' "

DMS deg, min, sec Useful for angle measurement.


Miscellaneous

P0, P1, P2

(x1, y1) subscripts

Used to denote multiple points or multiple variables where it is

inconvenient to use different letters

a → b it follows If Joe is older than Sam and Sam is older than Betty, it follows that

Joe is older than Betty. i.e. J > S, S > B → J > B.

a ↔ b if and only if

This is used for an implication that is valid in both directions.

Notice that the above example is not necessarily valid in the

opposite direction.

Greek Letters and their Common Use

α, alpha

unknown angle

γ, gamma

unknown angle

ρ, rho

density

θ, theta

angle from x-axis

φ, phi

angle from vertical

β, beta

unknown angle

τ, tau

torque

Δ, delta

change or shift

elevation

π, pi

ratio of circumference

to diameter

ω, omega

rotational velocity

1.1.2 Common Vocabulary Terms in Technical Mathematics

Here are some additional vocabulary terms and concepts. You should have seen most of these before

now. We will use many of these in this course though not necessarily in this first Chapter. Glance

through them and circle the ones you are not sure about. Then ask your instructor for further

clarification.

of as in '3/4 of' or '75 % of' is treated as multiplication.

e.g. ¾ of 12 = (¾)(12), 75% of 300 = (75%)(300)

the sign of a number + or − denotes the sign of a number. If omitted, the number is positive. 'Negative

(of)' or 'opposite (of)' changes the sign of a number. When written, the sign of a

number is the same symbol as addition or subtraction. However, most calculators use

different symbols (keys) for subtraction vs. negation.

e.g. negative (of) 10 = -10, opposite of -12 = 12.

division by zero Division by zero is undefined. It is an illegitimate calculation.

It can be denoted by DNE (does not exist), the empty set, , or simply undefined

The null set also refers to the empty set.

average The average of a set of numbers is their sum divided by the quantity of numbers in

the set. e.g. The average of { 5, 7, 12} = (5 + 7 + 12) / 3 = 8

ratio A ratio is an ordered pair of numbers used to show a comparison between like or

unlike quantities. It can be written as 'a to b', a:b or a/b.

Two ratios are equivalent ratios if they can be reduced to the same fraction.

proportion A proportion is an equation comparing two ratios.

numerator The numerator is the expression in the upper half of a fraction.

denominator The denominator is the expression in the lower half of a fraction.

mixed number A mixed number is a combination of a whole number and a fraction. e.g. 3 ½

improper fraction An improper fraction has a larger numerator than denominator. e.g. 9/5.

reduce A fraction is completely reduced when it uses the smallest possible whole numbers to

express an equivalent fraction. e.g. 12/15 is reduced to 4/5.


percent Percent literally means per 100. e.g. 20% = 20/100

exponentiation Exponentiation represents repeated multiplication of a base number.

Format: baseexponent

The exponent (or power) represents the multiplicity.

e.g. 32 = (3)(3). 4

5 = (4) (4) (4) (4) (4).

The second power is also called the square and the third power is also called the

cube. e.g. 5 squared = 52, 8 cubed = 8

3.

Another common expression is 'a to the b' (ab). e.g. four to the fifth = 4

5.

root, radical The square root of x ( x ) is the positive number, that when squared, equals x.

e.g. 25 = 5. is called the radical symbol and the 'x' is called the radicand.

We also say radical x to refer to x .

The cube root is the number, that when cubed, equals x. e.g. 3

8 = 2.

magnitude Magnitude denotes the size of a quantity without reference to sign. It often refers to

an estimate or rounded number.

place holder Place holders can be used to create fractions. e.g. (⅔)(5) = ( 2

3 )(

5

1 ).

The also refer to the zeros used to denote large or small numbers.

e.g. 7 million = 7,000,000. or 5 thousandths = 0.005.

(arithmetic) operation Arithmetic operations are combinations of negation, addition, subtraction,

multiplication, division, exponentiation and roots.

order of operations The order of arithmetic operations must obey an established hierarchy.

expression A general term used to refer to single numbers or variables as well as complex

combinations of numbers and variables. A mathematical 'phrase'.

equation An equation is a mathematical sentence equating two expressions.

variable A symbol denoting a quantity that can vary.

parameter A symbol denoting an unknown quantity that is predetermined.

whole number Whole numbers are the set { 0, 1, 2, 3, …}.

integer Integers are the set { … -3, -2, -1, 0, 1, 2, 3 …}

rational number Any number that can be represented by a fraction or repeating decimal.

irrational number Any number that cannot be represented by a fraction or repeating decimal.

e.g. π, 2

real number Any number that conforms to a real measurement.

(algebraic) factor A factor is an expression that multiplies another expression.

e.g. In (2)(3)(x + 5), (2), (3) and (x + 5) are factors.

(algebraic) term A term is an expression that is added to another expression.

e.g. In 4 + x + 5(2 − y), the 4, x and 5(2 − y) are terms.

numeric format An expression with any variable or parameter changed to its current value.

algebraic format An expression containing unknown variables or parameters.

1-D, 2-D, 3-D Referring to the space in which we are working. 1-D (one-dimensional) has only

length without width or depth. 2-D (two-dimensional) has length and width but not

depth. 3-D (three-dimensional) has length, width and depth.

plane figure A 2-D figure. Squares, rectangles, triangles and circles are common plane figures.


circumference The perimeter of a circle.

chord A line segment that connects two points on the circumference of a circle. The

diameter is special case of a chord.

right triangle A triangle with a 90° angle.

hypotenuse The side opposite the 90° angle in a right triangle.

trapezoid A 4-sided plane figure with two parallel sides.

parallelogram A 4-sided plane figure where both pairs of opposite sides are parallel.

polygon A many sided plane figure. Triangles, rectangles, pentagons and hexagons are

examples of polygons.

polyhedron A many faced 3-D figure. Cubes, tetrahedrons and pyramids are examples of

polyhedrons.

1.1.3 Number Classification System

We classify number into various categories. From early childhood we learn about whole numbers. The

whole numbers are the set { 0, 1, 2, 3 . . . }.

Integers are positive and negative whole numbers. i.e. { . . . -3, -2, -1, 0, 1, 2, 3 . . . }

Rational Numbers are numbers which may be written as a fraction or a repeating decimal. Rational

numbers can be positive or negative. Recall: Any decimal may be written as a fraction (e.g. 1.25 = 125

100 )

and any whole number may be written as a fraction (e.g. 7 = 7

1 ). So,

89

13 , -4.32, 0, 1,

3

4 , 7.5,

24367457

3658747665

are rational.

Irrational Numbers are those numbers which associate with a real measurement but still cannot be

exactly represented by a fraction or repeating decimal. Rational numbers can be positive or negative.

For example, π feet is the distance around a circle with a one foot diameter. Thus, the number π exists

as a real measurement.

And, 2 is the diagonal distance across a 1 x 1 square. Thus, the number 2 exists as a measurement in

the real world.

These values are real measurements although neither of these may be exactly represented by a fraction

or a repeating decimal. They are examples of irrational numbers.

Real Numbers are all possible measurements. In short, every real is either a rational or an irrational

number. The number line is a graphical representation of all the real numbers.

reals

irrationalsrationals

integers

e 32 3 2 6 p1120-1 -1.3 -2 -3 -/4

Another way of describing an irrational number is a decimal number that never repeats a sequence and

never ends. There are infinitely many rational numbers and infinitely many irrational numbers.


Furthermore, between any two rational numbers we can squeeze infinitely many irrational numbers.

Likewise, between any two rational numbers we can squeeze infinitely many irrational numbers.

However, there are considerably more irrational numbers than rational numbers. That is, there are

different degrees of infinity. If you find such ideas fascinating then you should consider pursuing

mathematics beyond this course.

Quite often we prefer numbers as exact rational numbers rather than approximate decimals. For

instance, 6⅛" vs. 6.13" or 2⅓ ft vs. 2.33 ft.

When an exact integer is given it is best not to use a decimal form since this implies an approximation

occurred to obtain the integer. For instance, since 4.003 ≈ 4.00 when we write 4.00 we assume the

actual number was somewhere between 3.995 and 4.005 and was simply rounded to 4.00. Thus, when

we have exactly 4 items we should just write 4 not 4.00. Unfortunately, just because we write "4" does

not automatically imply the number is exact. See Rounding § 1.9.

1.1 Exercises - Math Notation and Vocabulary

Choose true (T) if always true; otherwise choose false (F).

1) (a) T F 3(4 + 5) = 3[4 + 5] (b) T F 2{ 3 + 4[4 + (2)(9)] } = (2)(91)

2) (a) T F π ≈ 3.14 (b) T F 3.14 < π < 3.15

3) (a) T F 3.5 > 7/2 (b) T F 0.001 > 0.0002

4) (a) T F 93 = 999 (b) T F 5

4 = (5)(5)(5)(5)

5) (a) T F 3/5 = 3

5 (b) T F 3/5 = 0.6

6) (a) T F 4/5 = 80% (b) T F 0/10 = 10/0

7) (a) T F 8 – (2)(4)

(2)(3) – 6 = Ø (b) T F φ = Ø

8) (a) T F θ = Ø (b) T F Q2 = Q2

9) (a) T F 12 ft2 = 12 sq-ft (b) T F 8 '

2 = 8 sq-ft

10) (a) T F abc = θ a b

c

d

<abc ~

(b) T F PQ RQ P

Q R

PQ RQ

11) Write π accurate to the 5

th decimal place.

12) (a) Give an alternate designation for θ. θ =

(b) Mark "x" at abc

ab

c

d

13) (a) Fill in the blanks: _____ _____.

(b) Give an appropriate label for the right angle’s location.

P

Q R

PQ RQ

14) In the expression 12 3

4

(a) "3" is called the .

(b) "4" is called the .

(c) We call this a number.


15) Neatly draw a (a) trapezoid (b) parallelogram

16) In algebraic form write and expression for:

(a) the quotient of input to output using in & out.

(b) the ratio of lead to zinc using L & Z.

(c) the product of length and width using L & W.

(d) the average of three scores using T1, T2, T3.

(e) the difference of P and Q using P and Q.

(f) the difference between the squares of a and b.

17) Circle the integers: 6 -2 0 36 9

5 48

16 4.5 6 ½

18) Circle the real numbers: 9

5 6 ½ 16 − 4

2

12 − (3)(4) 37 -8 2

19) Circle:

(a) the first term 4x2 + 3x + 2 + 5y

2 + 3y

(b) the constant term 4x2 + 3x + 2 + 5y

2 + 3y

(c) the y2 term 4x

2 + 3x + 2 + 5y

2 + 3y

(d) the second factor 3 · x · (x2 + 2) · (2y – 7)

20) Label the diagram with these items.

(a) origin

(b) x-axis

(c) y-axis

(d) (3, 4)

(e) (-2, -5)

(f) (-5, 0)

(g) Quadrant I

21) Explain the difference between a mathematical expression versus a mathematical equation.

22) Use the following set to answer a, b & c: { 7, -3, 3/5, 3 , 0, 0.125, -2¾, 3.83¯¯, π, π/2 }

(a) List the integers (b) List the rational numbers (c) List the irrational numbers

23) Write the following as a mathematical equation.

(a) The difference between x and y is 7. (b) The sum of x and y is 15.

(c) The product of x and y is 21. (d) The quotient of x and y is 0.04

24) Place the proper inequality ( < , > ) in the blank.

(a) 9.009 ___ 3.999 (b) -3.7 ___ -3.6 (c) π ___ 3.14 (d) 37/54 ___ 28/35

25) Explain what is wrong with each of the following expressions

(a) {[(1 + 2)4] + 7(5 + 6)]8}9 (c) 65.9923.55 (d) 5" × 5" = 25 "2


1.2 Rules for Real Number Arithmetic

We want to be able to perform addition, subtraction, multiplication, division, exponentiation, take roots

and so on with real numbers. Assuming we recall how to handle the arithmetic of positive numbers we

shall here concentrate on operations with signed numbers (positive and negative).

In order to make working with positive and negative numbers easier, we need to find some way to make

sense of the ideas. Memorization is simply not a realistic approach. We should relate positive and

negative numbers to something that we already know.

Some people like to think of positive and negative numbers as numbers on the number line. Addition

and subtraction become moving to the right or the left on the number line. If that is an image that works

for you, use it! There are numerous examples of positive and negative numbers in real life, such as

temperatures, elevations, account balances, earning / spending, gambling results, etc. Use these

examples to help yourself visualize and understand the concepts of positive and negative numbers.

1.2.1 A Few Writing Conventions

"-a" is interpreted as the opposite of a or the negative of a. We often say 'negative a' for the written

expression '-a'. 'Negative five plus negative two' would refer to (-5) + (-2). Although parentheses are not

necessary when writing negative numbers we often write that way just to emphasize their occurrence.

We can always exchange (-a) ↔ -a or (+a) ↔ +a or +a ↔ a. Thus, we might choose to rewrite:

-5 + -2 ↔ (-5) + (-2) or -3 + -4 ↔ (-3) + (-4)

When a number is positive we seldom emphasize that fact with parentheses so (+a) is usually just

written as 'a'. The obvious exception is multiplication such as 7 · 3 ↔ (7)(3). Why is that?

We can also exchange: - (+a) ↔ - (a) ↔ - a ↔ + -a ↔ + (-a)

Since, the opposite of the opposite of a should be no change for 'a' we have: -(-a) ↔ - - a ↔ a.

Beware!

We can always exchange (-a) ↔ -a. But, the parentheses are redundant only when they surround a single

term. For example, -(a + b) ≠ -a + b i.e. -(2 + 3) ≠ -2 + 3

1.2.2 Using Money ($) to Understand the Arithmetic of Positive and Negative Numbers

For many people, thinking about positive and negative number arithmetic in terms of a bank account is

an easy way to make sense of things. Here we'll use the words credit (positive balance or receive $) and

debit (negative balance or pay $).

The opposite of a credit is a debit (and visa versa) so defining credits as (+) indicates debits are (−). The

opposite of a debit, -(-a), must be a credit, +a. So, -(-a) must be the same as +a. This fundamental

property of signed numbers is extremely handy: -(-a) = +a e.g. -(-6) = 6.

Now let's use these ideas to formulate the common sense rules for the arithmetic of signed numbers. For

example, making a deposit is akin to adding a positive number. Writing a check is equivalent to adding

a debit (adding a negative number) to the account. Withdrawing from the account is the same as taking

the money away or subtracting it from the account. And so on…


Adding Positive and Negative Numbers Case (-a) + (-b)

e.g. (-20) + (-30)

We write consecutive checks; one for $20 and one for $30. What is the result? We could

have written one check for $50. Thus we should combine the amounts and make the result

negative, (-a) + (-b) = -(a + b). e.g. (-20) + (-30) = -(20 + 30) = -50.

Case a + (-b)

e.g. 20 + (-5)

We deposit $20 then write a check for $5. How do we balance our account? We begin with

the deposit then subtract $5 to obtain the balance. We turn the addition into a subtraction,

a + (-b) = a − b. e.g. 20 + (-5) = 20 − 5 = 15.

Case a + (-b)

e.g. 20 + (-25)

We deposit $20 then write a check for $25. What is the result? We are overdrawn by the

difference between the account and the check amount. We subtract the amounts in reverse

order (to obtain their difference) but make the result negative (overdrawn), a + (-b) = -(b − a).

e.g. 20 + (-25) = overdrawn by the difference = -(25 − 20) = -5.

Case (-a) + b

e.g. (-8) + 20

We write a check for $8 but quickly deposit $20 to cover the check. What is the result? The

deposit was enough to cover the check so we can treat this the same as if we had started with

$20 in the account and then wrote the $8 check. Thus, we turn the addition into a subtraction,

(-a) + b = b − a. e.g. (-8) + 20 = 20 − 8 = 12.

Case (-a) + b

e.g. (-20) + 18

We write a check for $20 and quickly deposit $18 to try to cover the check. What is the

result? The deposit was not enough to cover the check so we are again overdrawn by the

difference between the deposit and the check amount. Thus, we should subtract the amounts

(to obtain their difference) and make the result negative (overdrawn), (-a) + b = -(a − b). e.g.

(-20) + 18 = overdrawn by difference = -(20 − 18) = -2.

Subtracting Positive and Negative Numbers

Subtracting a positive number should be exactly the same as adding a negative number. Why is that?

Consider two choices for your account: Write a check for $10 (add an IOU of $10 to the account) or

withdraw $10 directly from the account (subtract $10). In our account metaphor, they both get

interpreted as reductions to the account (a debit). Acct. Balance + (-$10) = Acct. Balance − $10.

Thus, every subtraction can be made into addition a − b = a + (-b) and every addition can be made into

subtraction. a + b = a − (-b)

With the above logic we can turn each subtraction into an addition to resolve the following cases.

Case a − b

e.g. 3 − 7

Deposit $3 then withdraw $7. It's the same as depositing $3 and then writing a check for $7:

Either way, we're overdrawn by the difference of $4; a − b = a + (-b) = -(b − a).

e.g. 3 − 7 = 3 + (-7) = overdrawn by the difference = -(7 − 3) = -4.

Case (-a) − b

e.g. (-2) − 9

Write a check for $2 then withdraw another $9. It's the same as writing a check for $2 and

then writing another check for $9: We're overdrawn by $11. We combine the two negatives;

(-a) − b = (-a) + (-b) = -(a + b). e.g. (-2) − 9 = (-2) + (-9) = -(2 + 9) = -11

Case a − (-b)

e.g. 100 − (-50)

Subtracting a negative number, what could that mean? Suppose we

have a +$100 balance in one account and -$50 in another account.

What is their difference? We end up with a difference of $150. It

turns out to be exactly the same as if we added $100 + $50. Thus,

subtracting a negative becomes adding a positive; a − (-b) = a + b.

e.g. 100 − (-50) = 100 + 50 = 150. fig 1


Summary of Addition and Subtraction with Signed Numbers (Let a, b ≥ 0) Case: a + b Perform normal addition. 12 + 7 = 19

Case: a – b When a ≥ b, perform normal subtraction. 10 − 6 = 4

Case a − b When b > a, use a – b = -(b – a). e.g. 18 – 23 = -(23 – 18) = -(5) = -5

Case: a + (-b) Use a + (-b) = a – b. e.g. 9 + (-8) = 9–8 = 1, 7+(-9) = 7–9 = -(9–7) = -(2) = -2

Case: (-a) + b Use (-a) + b = b + (-a). e.g. -7 + 3 = 3 + (-7) = 3 – 7 = -(7 – 3) = -(4) = -4

Case: (-a) + (-b) Use -a + (-b) = -(a + b). e .g. -7 + (-8) = -(7 + 8) = -(15) = -15

Case: (-a) – (b) Use -a – (b) = (-a) + (-b). e.g. -9 – (2) = -9 + (-2) = -(9 + 2) = -11

Case: a – (-b) Use a – (-b) = a + b. e.g. 17 – (-12) = 17 + 12 = 29

Case: (-a) – (-b) Use -a – (-b) = (-a) + b. e.g. -7 – (-9) = -7 + 9 = 9 + (-7) = 9 − 7 = 2

In general, we can rewrite any subtraction as an addition when the signs are the same. We add the unsigned

numbers and give the sum the common sign. When the signs are different, we find the difference of the

unsigned numbers and use the sign of the larger number. Review the accounting examples until this makes

complete sense. Do not simply try to memorize these rules!

Note: When adding or subtracting, any pair of adjacent + & − signs may be combined into one.

In general: + – ↔ – – + ↔ – + + ↔ + – – ↔ +

Example 1

a) 8 + (-3) = 5 b) (-4) + (-3) = -7 c) (-5) + (3) = -2

d) 21 − 27 = -6 e) (-14) + 23 = 9 f) 22 + (-43) = -21

g) (-23) − (-28)= 5 h) 17 − (-8) = 25 i) (-18) − 23 = -41

j) -(-7) = 7 k) -[-(6)] = 6 l) -[-(-8)] = -8

Multiplying Positive and Negative Numbers One way to examine multiplication and division of signed

numbers is to consider a multiplication table. Fill in the times-

table cells using the rules we have for positive numbers. For

example, 2 × 3 = 6.

Our final result must also be consistent with how we interpret

multiplication. For instance: (1)(a) = a. So, 1(-3) = -3, etc.

Also, (2)(-1) = (-1) + (-1) = -2. Fill in these cases.

Now, look for patterns and fill in any remaining cells making

sure the entire pattern is consistent.

× -b -3 -2 -1 0 +1 +2 +3 +b

+a 0

+3 0

+2 -2 0 6

+1 -3 0

0 0 0 0 0 0 0 0 0 0

-1 0

-2 0

-3 0

-a 0

fig 2

You should notice that in general:

(+a)(+b) = +ab (+a)(-b) = -ab (-a)(+b) = -ab (-a)(-b) = +ab


Note: We can also use a similar kind of money reasoning as we used for addition and subtraction.

(4)(-20) Suppose we have written four $20 checks. Then we would have a total debit of $80.

Thus, (4)(-20) = -80 or (a)(-b) = -a·b

(-2)(-500)

If we payout $500 twice, that would be 2·(-$500) = -$1,000. But, (-2)(-500) should

mean the opposite of paying $500 twice. That is it should mean we receive $500 twice.

Hence (-2)(-500) = -[2(-500)] = -[-1,000] = +1,000.

-12

3

Suppose we are overdrawn by $12 and we split that equally among three separate

accounts. Each account would be overdrawn by $4. Thus, -12/3 = -4

We can create other ways to consider the logic of multiplication and division with signed numbers but in

general, when we multiply (or divide) any two positive or negative numbers:

Multiplication of Positive and Negative Numbers

If the signs are the same, the answer is positive.

If the signs are different, the answer is negative.

× -c +d

-a +a·c -a·d

+b -b·c +b·d

Division of Positive and Negative Numbers

Every multiplication problem is interchangeable with a division problem. For example, half of 6 cups is

the same as 6 cups divided in two. i.e. 1

2 · 6 cups = 6 cups ÷

2

1 . Thus, multiplication and division must

both obey the same rules regarding signed numbers. Since 1

2 · (-6) = -3, also

-6

2 = -3. Thus,

Division of Positive and Negative Numbers

If the signs are the same, the answer is positive. +a

+b =

a

b

-a

-b =

a

b

If the signs are different, the answer is negative. -a

+b = -

a

b

+a

-b = -

a

b

Repeated Operations of Multiplication and Division

Since any pair of multiplication and/or division operations involving negative numbers can be simplified

to a positive, the sign of the final calculation will always be negative if there are an odd number of

negative numbers involved in the calculation and the sign of the final calculation will always be positive

if there are an even number of negative numbers involved in the calculation.

Example 2

a) (8)(-3) = -24 b) (-4)(-3) = 12 c) (-5)(3) = -15 d) (-3)(-4)(-2) = (12)(-2) = -24

e) (4)(-3)

(-2)(-6) = -1 f) (-4) + (-3) = -7 g) -5 + 13 = 8 h)

(-3)(-4)(5)

(3)(4)(-5) = -1

i) -4

2 = -2 j)

-9

-3 = 3 k)

20

-4 = -5 l)

-1

2 =

1

-2 = -

1

2 = - 0.5


Notice in the last example that placement of the negative sign is somewhat optional when writing a

fraction. However, by convention we usually place these negative signs either in the numerator or

directly preceding the fraction. This is the same whether we write with a horizontal or diagonal divider;

e.g. - 1

2 = - ½.

Beware! We seldom leave multiple negative signs where reduction to a single sign is possible but be careful not to

mistakenly cancel negatives that are not directly adjacent to each other. The following examples are common

mistakes.

-8 + 5

-2 ≠ 8 +

5

2 (-2) + (-3)(4) ≠ 2 + (3)(4)

-1

2 +

1

-2 ≠

1

2 +

1

2

We can only cancel "pairs of negatives" when the entire expression consists exclusively of multiplication and

division factors. If multiplication (or division) is mixed with addition or subtraction we cannot perform any

shortcut cancellations.

(8)(-2)

(5)(-2) =

(8)(-2/ )

(5)(-2/ )

7 + (-5)

(-5) ≠

7 + (-5/ )

(-5/ )

5 + (-2)

3 + (-2) ≠

5 + (-2/ )

3 + (-2/ )

-(-a) = +a only when the negative and subtraction are directly adjacent to each other or the parentheses

contains only factors. e.g. -(-3·π) = 3·π but -(-3 + π) ≠ 3·+ π

Exponents with Positive and Negative Numbers

Exponentiation is equivalent to repeated multiplication. Since we now know how to multiply signed

numbers we should be able to calculate the result of positive or negative numbers which are raised to a

power. e.g. (-5)3 = (-5)(-5)(-5) = -125. In general, a negative number taken to an odd power will have a

negative answer and a negative number taken to an even power will have a positive answer. Why?

Example 3

(-1)even power

= 1 (-1)odd power

= -1 (-1)12

= 1 (-1)7 = -1

(-5)2 = (-5)(-5) = 25 (-4)

3 = (-4)(-4)(-4) = -64 (-½)

3 = (-½)(-½)(-½) = -⅛ -2

2 = -(2·2) = -4

Beware! Watch out for expressions like -2

2. Perhaps it's not clear whether this is to be interpreted as the opposite of

two squared or negative two repeated twice. In fact,

-22 is always interpreted as the opposite of 2

2 = -4.

(-2)2 is always interpreted as negative two repeated twice (-2)

4 = (-2)(-2) = 4.

See Order of Operations § 1.2.3 for further clarification.

The usual question that arises for the curious of heart is how should we interpret negative numbers in the

exponent? The answer to that and related questions is discussed later in the chapter.


Practice Try performing these calculations without using a calculator. 1) (a) 13 – 18 = (b) -18 – (-12) = (c) 7 + (-11) =

2) -3 + 5 = -(-4) = -15 – 27 =

3) -6 – 13 = (7)(-16) = 16 + (-5) =

4) (5)(-3) = (-8)(-7) = 8 – 11 =

5) -15

30 =

-18

-24 =

(-2)(4)(-3)

(2)(3)(-5) =

6) (-2) – (-7) = -(-4) – (-27) = (-24) + (-12) =

7) -(-6) = (-7)(-1)(6) = 16 + (-25) =

8) (-5)(-3) = -(-8)(-7) = 28 – 31 =

9) -45

-30 =

-8

-24 =

(-5)(-4)(-3)

(-1)(-2)(-3) =

10) -45 = -(-4)

3 = (-2)

4 + (-1)

2 =

Answers-

1) -5; -6; -4

4) -15; 56; -3

7) 6; 42; -9

10) -1024; 64; 17

2) 2; 4; -42

5) -1/2; 3/4; -4/5

8) 15; -56; -3

3) -19; -112; 11

6) 5; 31; -36

9) 3/2; 1/3; 10

1.2.3 Order of Operations

Order of Operations are the rules that give the conventions (or precedence) when multiple operations

occur in a single expression. For instance, in the expression 13 – 3 · 2 we get a different result

depending on whether we first multiply or first subtract. Prove it! To keep mathematics consistent we

require a standardized procedure when multiple operations occur and following these conventions

precisely eliminates any ambiguities in a calculation.

In general, we do the more sophisticated operations first. However, this is not a very precise definition.

More precisely, the order of operations must obey the following priorities:

Highest Priority (do these first)

Parentheses

&

Groupings

Parentheses override all other operations. When nested " { [ ( ) ] } ", the inner

parentheses have highest priority and when sequential " [ ] [ ] [ ] ", the left most

parentheses have highest priority. Brackets, extended fraction dividers, radicals and all

other grouping symbols infer parentheses and thus have the same priority as

parentheses.

Exponents

&

Roots

If exponents are stacked " 234

", the upper-most pair has highest priority. If exponents

are sequential " 23 + 3

2 – 4

3 ", the left-most exponent has highest priority. An

individual root is equivalent to an exponent and the root of an expression implies

parentheses since it is a grouping symbol. See §

Multiplication

&

Division

These share the same priority level. As before, with multiple operations the left-most

operation has highest priority. e. g. 3 · 8 ÷ 4 · 2. Work from left to right


Addition

&

Subtraction

These share the same priority level. As before, with multiple operations the left-most

operation has highest priority. e. g. 3 + 8 – 4 + 2. Work from left to right.

Negation Though equivalent to subtraction in priority there is the common problem of correctly

interpreting -ab or -(expression). Note! -a

b = -(a

b) ≠ (-a)

b.

Lowest Priority (do these last)

How do we remember this order?

This is one of the few times we must resort to memorization. Some people remember the mnemonic

PEMDAS while some people remember the sentence, Please Excuse My Dear Aunt Sally. Some

people make up their own memorable sentences that have the same initial letters.

Before we practice order of operations let's take a look at some places where mistakes commonly occur.

Perhaps forewarned will be forearmed!

Beware These Common Pitfalls! The calculation 25 − 15 × 43 tempts us to perform the subtraction first because that calculation is enticingly

simple and it comes first. Such pitfalls can be avoided by writing 25 − (15 × 43). Although the parentheses

are redundant they help eliminate a potential error.

The calculation 123 × 25 − 15 might also tempt us to perform the subtraction first because the calculation is

enticingly simple compared to the multiplication. As before, consider including redundant parentheses, e.g.

(123 × 25) − 15, to help eliminate a potential error.

Another common error seems to occur when the order of operations is

out of sync with the left-to-right priority. Remember, the operational

priority take precedence. The left-to-right priority only comes into play

with multiple operations all having the same priority.

e. g. 3 + 4 + 5 × 6

It’s tempting to work left to right but

the multiplication operation 5 × 6

has priority!

Watch out for other grouping symbols besides parentheses.

Fraction dividers and radicals are a grouping which imply

parentheses. Since we so often use a calculator to key in

such expressions it is imperative to understand the order of

operations (PEMDAS).

expression

expression → (expression)/(expression)

expression → (expression)

When in doubt use extra parentheses to remove any chance of ambiguity and to help the reader avoid

mistakes. This is especially important if you plan to use a calculator. But be careful, parentheses can change

the value of an expression if they change the order of operations.

4

2 + 5 = 4/(2 + 5) = 4/7

4

2 + 5 ≠ 4/2 + 5 = 7

4

5π = 4/(5π)

4

5π ≠ 4/5π

4 + 5 = (4 + 5) = 9 = 3 4 + 5 ≠ 4 + 5= 2 + 5

What is the difference between (-a)n vs. -a

n ? Notice that parentheses make a difference.

(-3)2 = (-3)(-3) = 9 these are different! -3

2 = -(3 · 3) = -9

-32 ≠ (-3)(-3) this is a very common mistake!


Practice with Order of Operations (Try these without using a calculator)

1) a) 4 · 5 – 3 = b) 52 – 3

2 = c) (3 + 2)·10 – 5 =

2) a) 6 + 9 + 10

2 + 3 + 5 = b)

(6 – 4)(5 – 3)

6 + 4 – 2 · 5 = c)

6 · 15

25 · 9 =

3) a) 10 ÷ 12 ÷ 1

8 ÷

5

9 = b)

16

4

8 =

c)

16

4

8

=

4) a) -22 = b) 3

2 + 4

2 = c) [28 − 4]

2 =

5) a) 16 – 3 · 5 + 5 = b) 4 + 8

8 – 6 = c) 8(9 – 6)10 – 5 =

6) a) 0 ÷ 10 = b) 3 · 10 – 10 · 2 = c) (3 · 5)2 + 3 · 5

2 =

Answers-

1) 17, 16, 45

2) 5/2, Ø, 2/5

3) 12, 1/2, 32

4) -4, 5, 576

5) 6, 6, 235

6) 0, 10, 300

1.2.4 Applied Order of Operations

Writing an accurate expression in terms of order of operations from verbal or written directions can be

tricky. Placement of commas or verbal pauses or inflection can alter the meaning of an expression. If

you are not completely sure of the speaker’s intent, it would be wise to ask for clarification.

Example 1 Write a valid mathematical expression for each of the following statements

1)

Bill must order sod for twelve rectangles each

measuring thirty six feet by ninety two feet. What

amount of sod is needed?

Sod = 12 ( L× W)

12 (36' × 92') = 39,744 sq-ft

2) Four laborers each earn $7.85/hr while two supervisors

each earn $14.50/hr. What is the payroll for 40 hrs?

40 hr (4 × $7.85/hr + 2 × $14.50/hr) =

$2,416

3)

The area of each rectangular face is

found using its length × width.

What is the formula for the total

surface area of this open box?

A = 2(L × H) + 2(W × H) + L × W =

2LH + 2WH + LW

4)

This trapezoid’s area is found multiplying

the average length times its height. What

is the formula for the total area?

A = a + b

2 × c

1.2 Exercises- Order of Operations

Simplify to an exact integer or a reduced fraction. Do not use a calculator.

1) 2 · 8 − 4 2) 3 · 10 − 10 3) 12 · 8 ÷ 4

4) 1 + 9 · 10 5) 2 + 8 · 8 + 2 6) 10 − 5 (10 − 5)

7) -32 8) 2

3 · 3

2 9) (4 − 2)

2

10) 4(2 − 2) 11) 4 · 3 − 2 · 6 12) 2 · 12.1 − 2.1


13) 9[7 − 2 + 6] 14) 3.2 · 6.5 + 3.5 15) 12 ÷ 4 ÷ 3

16) 24 ÷ 12 ÷ 6 · 3 17) 21 − 4 · 5 − 5 18) 1000 ÷ 10 ÷ 10 ÷ 10

19) 10 ÷ 0 20) 101 − 10 ÷ 10 21) 0 ÷ 20

22) 4 + 8

8 − 4 23)

4 + 8

8 − 2 · 4 24)

3 · 4 − 6 · 2

8 − 4

25) 5 + 5[50 − 5(10 − 5)] 26) 2+2{2+2[2+2(2+2)]} 27)

1 + 1

1 + 1

1 + 1

1 + 1

28) 13 – 18 29) -18 – (-12) 30) 7 + (-11)

31) -3 + 5 + (-2) – (-7) 32) -(-4) – 3 + (-12) – (-27) 33) -15 – 27

34) -2(6 – 13) 35) -5 (7 + (-16)) 36) -11 + 13 – 16 + (-5) + 8

37) -(5)(-3) 38) (2 – 8)(-7) 39) (8 – 11)(-5 – 2)

40) -15

24 41)

-18

-24 42)

-2(4)(-3)

2(-3)(5)

43) -2(-3) + (-2)

3 + (-3)(5) 44)

-2(-4) + (-3)3

5 + (-3)(2) 45)

-2(4) + (-3)

-7 + (-3)(5)

46) 3(4) – 12

2 – 2 47)

(-3) + 2(-6)

3 + 8(-6) 48)

3(-2) + 6

37(-2)

49) (-4)3 50) [-3(5–7)+(-2)]6÷16–15 51) 33 – 3

3

52) (-4)3 – 4·3

4 53) [-3(5 – 8)+4]4÷13−26 54) -2

4 · 4

2

55) -15 ÷ 3 56) [5(2–5)+3]2÷(-8)–(-6) 57) 23 – 32

4·6 – 52

58) 2

3×103 59) 200 − 100[27 − 32] 60) [696 − 701]400 − 300

61) 1 + 23 − 3 62) (2 – 8) ÷ (-3) 63) (2 – 8)/2

64) 8 − 2(6 – 13) 65) 3 − 2(4 + 6) 66) 20 − 10(4 − 6)

67) (5)(-4) ÷ -10 68) 10 − 5 (2) 69) 24 − 4 (3)/6

70) 2(-3) + (-10)

3 + (-3)(-5) 71)

-2(-4) + (-3)3

5 + (-3)2 72)

-24 + (-3)

-7 + (-3)2

73) 3(4) – (-2)(-6)

9 – 2 74)

(-7) + (-2)(-6)

3 + 8(-6) 75)

3(-2) + 5 + 18

37(-2)

76) (32 + 4)

2 77) (2

3 − 3

2)

3 78) (-1)

27

Applied Order of Operations Exercises- Write a valid expression for each statement then compute the result

79) A 22' pipe has a 7' and 5'piece cut off. How

much remains?

80) A welder must weld three lengths together; 42"

37" and 16". How long is the resulting piece?

81) A hybrid car that gets 52 mpg (miles per gallon)

has a 12 gal gas tank. How many miles could

the car travel on one tank full of gas?

82) Sydne has 3 regular employees earning

$8.65/hr and a supervisor earning $13.25/hr.

What is the total payroll for a 40 hr week?

83) The following amounts of hydraulic fluid have

been recorded. What is the net result?

Inputs: 32 cc, 24 cc, 92 cc, 18cc

Outputs: 16cc, 27cc, 104cc, 19 cc.

84) A cyclist averages 3 m/s on a flat trail, 5 m/s on

a downhill and 1 m/sec on uphill. If she travels

120 m on a flat trail, 25 m of downhill and 50

m of uphill how long does this segment of her

journey take?


85) Kermin orders 1,256' of siding. If each piece is

12' long, how many pieces should be delivered?

86) A V8 has a total displacement of 2000 cu-in.

What is the displacement of each cylinder?

87) A plane can climb at 420 ft/min. How long will

it take the plane to gain 5200 ft?

88) A plane can climb at 420 ft/min. How far will

it climb in 15 min?

89) Paul has four boxes of washers with the

following quantities: 143, 162, 154, 137. What

is the average number of washers per box?

90) What is the formula for the average of the four

amounts; a, b, c, d?

91) A 17' wall is to have three 3' windows evenly

spaced (including the ends). How much space

is between each window?

92) A 28' wall is to have two picture windows

spaced 4' apart. What is the spacing on each

end of the wall if the windows are 5' wide?

93) A nurse administers 6 gm/hr of a sedative. How

much will be administered in 4 hrs?

94) A nurse administers 12 gm/hr of a drug. How

long until the patients has received 200 gms?

95) A young red maple tree watered at regular

intervals will grow in circumference by almost

5cm per year. If the planted tree is 11cm in

circumference when planted, how large is the

circumference of the maple after 6 years?

96) The average length of stay for a quest at a local

hotel is 2.5 days. The hotel averages 500

guests a month and a single night's stay costs a

guest 100 dollars. How much does the hotel

gross monthly?

97) John has a 10' × 10' wall section in which he

would like to center a 4' × 4' window. How far

from the ceiling is the top of the window?

98) A farmer harvests 232 bushels per acre on

average for his 240 acre parcel. He sells the

wheat for $2.38 per bushel. What is his return?

99) Joey & Jerri set aside $400 for a hiking trip to

the Wind Rivers with another couple and a

single friend. The food cost $310 but it was

split among the group of 5 as were the travel

expenses of $220. Misc. personal items cost

$35 each and misc. group items cost $160. How

much was left from Joey & Jerri's budget?

100) Suppose cats have 7 offspring per year on

average and it takes 12 months for a kitten to

reach reproductive maturity. There are 12 male

cats (8 of which are neutered) plus 10 female

cats (5 of which are neutered) on your block.

How many cats will be residing on your block

by the end of one year assuming none get

neutered, die or move out of the area?

101) Suppose 8" of snow (on average) is equal to

receiving 1" of rain in moisture. How many feet

of snow would the Ho Rain Forest get in a

really cold year if its annual rainfall is 180"?

102) A home requires 121' of sewer line which

comes in 10' sections. What size is the last

section of pipe?

103) Kevin’s home is listed for $869,900. He owes

creditors $670,000 for its 10 month remodel. If

the home sells for $799,900 how much did he

earn per month for his remodel?

104) Coffee costs $9 per pound. Antoine brews a

pound of coffee per week but only drinks about

a third of what he brews. How much money

does he waste each year in coffee expense?

105) A season ski pass is $966. Each round trip to

the mountain in your SUV costing $11 on

average. Hot chocolate and power bars run $5 a

day. If Betsy skis 42 days this year, what will

be her average cost for each day of skiing?

106) A season ski pass is $966 while a day pass

costs $53. Diana snowboarded 20 days last

year, but forgot her pass twice. Each

temporary ticket cost her $5. What was her

average cost per day boarding? Was it worth

buying the pass?

107) An electrician uses 2880 feet of 12-2 wire for a

job. The wire weighs 30 lb per 100 ft roll.

During installation, 360 feet is wasted and 100

feet of wire is stripped and thus looses half its

weight, how much weight in wire is the building

actually holding?

108) On average, Enu spends $12 dollars a week on

fresh vegetables and $8 dollars a week on fresh

fruit. Half the vegetables go to waste in his

fridge and one quarter of the fruit. How much

money does he 'throw away' each year?


1.3 Working with Fractions

This section on fractions is deliberately brief. It is intended as a review and quick reminder of things

that you should already know how to do. If you find that you need more help, try extra tutoring or you

may need to consider starting with a pre-algebra course instead.

When working on a “fractions problem”, try to think about why the procedures you're using works. If

you understand why a procedure works, you'll have a better chance of remembering how to do it!

Before we review procedures let's review the terminology and notation.

1.3.1 Notation and Vocabulary numerator

denominator or numerator / denominator

Note: When the numerator or denominator contains an expression then parentheses must usually be

used when writing the fraction with the "/" symbol as in 7 – 3

3 + 5 → (7 – 3)/(3 + 5) = 4/8 = 1/2.

In a mixed number there is a fractional part and a whole number part. The fractional part must be

between 0 and 1 as in 3 ¾.

In an improper fraction the numerator is larger than the denominator. We can convert a mixed number

to an improper fraction by adjusting the fraction to absorb the whole number part. 3 ¾ = 15

4 .

1.3.2 Equivalent Fractions

Fractions that represent the same (equivalent) amount are called

equivalent fractions. e.g. 1/2, 3/6 and 6/12 all represent exactly

half of the whole and are therefore equivalent fractions.

Geometrically, equivalent fractions cover equivalent areas. See

fig 1.

6/12 3/6 1/2

fig 1 Equivalent Fractions

A quick check to see whether two fractions are equivalent is to divide and find their decimal equivalents.

If their decimal equivalents match, then the fractions are equivalent.

Example 1 Show that 48

72 and

2

3 are equivalent fractions.

48

72 = 0.6666

_,

2

3 = 0.6666

_. Thus,

48

72 and

2

3 are equivalent fractions.

We can change any fraction into another equivalent fraction by multiplying (or dividing) both the

numerator and denominator of the fraction by the same number.

Example 2 Create three equivalent fractions of ⅔.

(a) 2

3 =

2·2

3·2 =

4

6 (b)

2

3 =

2·5

3·5 =

10

15 (c)

2

3 =

2·24

3·24 =

48

72

Thus, 2

3 ,

4

6 ,

10

15 and

48

72 are all equivalent fractions


We say that a fraction is reduced to its simplest form (is reduced or simplified), if the numerator and

denominator have no remaining factors in common. In other words, there is no equivalent fraction with

smaller integer in the numerator and denominator. Thus, the reduced or simplest form of 48

72 is

2

3 .

Example 3 Reduce the fraction.

(a) 6

10

6

10 =

3·2

5·2 =

3·2

5·2 =

3

5 (b) 9

4

16

4

16 =

1·4

4·4 =

1

4 → 9

4

16 = 9

1

4

1.3.3 Mixed Numbers and Improper Fractions

To change an improper fraction to a mixed number, perform long division, writing the remainder as a

fraction.

Example 4 Convert to a mixed number.

(a) 17

5

17

5 = 17 ÷ 5 → 3 with remainder 2

So, 17

5 = 3

2

5

(b) 19

3

19

3 = 19 ÷ 3 → 6 with remainder 1

So, 19

3 = 6

1

3

To change a mixed number to an improper fraction, multiply the whole number by the denominator then

add that product to the numerator. This becomes the new numerator. The denominator stays the same.

Why does this always work?

Example 5 Convert to an improper fraction.

(a) 3 2

5 5·3 = 15 so 3 =

15

3

So, 3 2

5 =

15

3 +

2

5 =

17

5

(b) 5 6

7 7·5 = 35 so 5 =

35

7

So, 5 6

7 =

35

7 +

6

7 =

41

7

1.3.4 Fractions and Decimals

To change from a fraction to a decimal, perform long division or use a calculator. This will often result

in an approximate decimal unless we indicate any repetition of decimal digits. We usually indicate digit

repetition by drawing a line over the repeating digits. For instance, 0.333… = 0.3̄, 5.373737… = 5.37¯¯.

Example 6 Convert to a decimal approximation.

(a) 17

5 → 17 ÷ 5 = 3.4 (b) 5

6

7 5 + 6 ÷ 7 = 5.957142… = 5.857142¯¯¯¯¯¯

To change a terminating decimal number to its fraction equivalent, carry the whole number and rewrite

the decimal portion as a fraction. Create the fraction according to the place value of the decimal then

reduce if necessary. For instance, W.x → W x

10 , W.xx → W

xx

100 , etc.

Example 7 Convert to a reduced fraction.

(a) 0.3 0.3 = 3

10 (b) 0.25 0.25 =

25

100 =

5·5

5·5·4 =

1

4

(c) 0.36 0.36 = 36

100 =

9·4

25·4 =

9

25 (d) 5.45 5.045 = 5 +

45

1000 = 5 +

9·5

200·5 = 5

9

200


To change from a repeating decimal to a fraction, put the digits that repeat over the same number of 9's

then reduce if necessary. Many calculators can convert a decimal to a fraction but be sure to enter

enough digits to adequately represent decimal repetition.

Example 8 Convert to a reduced fraction.

(a) 0.363636... 36¯¯ = 36

99 =

4 · 9

11 · 9 =

4

11 (b) 0.533... 0.53̄ =

5

10 +

1

10 3

9 =

1

2 +

1

30 =

15

30 +

1

30 =

16

30 =

8

15

Try this Outrageous Example: 0.142857142857142857... = (Answer 1/7)

Try this Challenge Example: 0.0836363636... = (Answer 23/275)

Any decimal number that neither terminates nor repeats cannot be written as a fraction. Such numbers

are called irrational numbers. π and 2 are examples of an irrational number.

1.3.5 Adding and Subtracting Fractions

To add or subtract fractions requires a common denominator (CD). You can only meaningfully add or

subtract "pieces of a pie," if the pieces are all the same size. We add the numerators keeping the same

common denominator. Can you explain to yourself why this makes sense?

Adding Fractions

+ =

5

12 +

7

12 =

12

12

5

12 +

7

12 =

5 + 7

12 =

12

12 = 1

Example 9

(a) 1

5 +

3

5 =

1 + 3

5 =

4

5 (b)

11

12 −

5

12 =

11 − 5

12 =

6

12 =

1

2

With mixed numbers we can either keep them as mixed numbers or convert them to improper fractions.

Example 10

(a) 7 2

3 + 5 2

3 = 7 + 2

3 + 5 + 2

3 = 12 + 4

3 = 12 + 11

3 = 13 1

3 (b) 7 2

3 + 5 2

3 = 23

3 + 17

3 = 23 + 17

3 = 40

3 = 13 1

3

Beware!

With negative mixed numbers or when subtracting mixed numbers it is very easy to make a sign

error. Remember, both the whole number and its fraction must have the same sign. In such

cases, it is much better to convert mixed numbers to improper fractions.

Example 11

(a) 7 3

8 − 5 5

8 = 59

8 − 45

8 = 59 − 45

8 = 14

8 = 7

4 = 1

3

4 (b) -4 1

8 − 3

3

8 =

-33

8 −

27

8 =

-33 + -27

8 =

-60

8 =

-15

2 = -7

1

2

If the fractions have different denominators we must first adjust one or both of the fractions to obtain a

common denominator. It is convenient to use the least common denominator (LCD) but any common


denominator will actually do. If we do not readily see the LCD we can obtain a common denominator

by simply multiplying the denominators.

Example 12

1

2 +

1

3 = Let CD = 2·3 (note LCD = 6)

1·3

2·3 +

1·2

3·2 = create equivalent fractions

3

6 +

2

6 = simplify the equivalent fractions

3 + 2

6 = add the fractions

5

6 simplify

Example 13

7

12 −

8

15 = Using the LCD = 60 = 15·4 = 12·5

7·5

12·5 −

8·4


35

60 −

32


35 − 32

60 =· add the fractions

3

60 =

1 3

20 3 =

1

20 reduce and simplify

Notice that using the LCD does not always eliminate having to reduce at the end. However, if you use a

larger common denominator than necessary you will always have to reduce at the end.

Example 14

3

15 +

7

12 = Let CD = 15·12 (note LCD = 60)

3·12

15·12 +

7·15


36

180 +

105


36 + 105

180 = add the fractions

141

180 =

3·47

3·60 =

47

60 reduce and simplify

1.3.6 Multiplying Fractions

Before multiplying fractions convert any mixed numbers into improper fractions. Then we multiply

straight across. e.g. 3

4 ·

4

5 =

3 · 4

4 · 5 =

12

20 =

3

5 Can you explain to yourself why this always works?


Multiplying Fractions

x =

3

4 ×

4

5 =

12

20

3

4 ×

4

5 =

3 × 4

4 × 5 =

12

20 =

3

5

Note: We can reduce the resulting fraction to its simplest form before or after we've multiplied, but it's

easier to cancel any common factors from the top and bottom before we multiply.

Example 15

(a) 3

8 ·

20

21 =

3

2·4 ·

4·5

3·7 =

5

2 · 7 =

5

14 (b)

5

8 ·

4

5 =

5 · 4

8 · 5 =

5 · 4

2 · 4 · 5 =

1

2

(c) 1 1

4 · 5 2

3 = 5

4 ·

17

3 =

5 · 17

4 · 3 =

85

12 = 7

1

12 (d) 1

1

2 · 2

2

3 =

3

2 ·

8

3 =

8

2 = 4

Beware!

We can only cancel factors against factors. DO NOT cancel factors against terms.

CORRECT 3 + 8

8 =

11

8 INCORRECT

3 + 8

8 INCORRECT

a + b

b

1.3.7 Dividing Fractions

8 divided in two is represented by 8 ÷ 2

1 while the equivalent concept half of 8 is represented by 8 ×

1

2 .

That is, dividing by 2

1 is equivalent to multiplying by 1

2 . If we considered enough similar examples we

would see a clear pattern. Every division operation has an equivalent multiplication operation. Namely,

division is interchangeable with multiplication by reciprocating the divisor. That is, Q ÷ a

b ↔ Q × b

a .

To divide fractions, first convert any mixed numbers into improper fractions. Then reciprocate (flip) the

dividing fraction and multiply instead. Can you explain to yourself why this always works?

Example 16

(a) 3

4 ÷

3

8 =

3

4 ·

8

3 =

3·8

4·3 =

2

1 = 2 (b)

4

15 ÷

1

3 =

4

15 ·

3

1 =

4 · 3

15 · 1 =

4

5

(c) 3 ¾ ÷ 2 ⅝ =

15

4 ÷

21

8 =

15

4 ·

8

21 =

15 · 8

4 · 21 =

10

7 = 1

3

7 (d)

8 2

3

3 5

9

=

26

3

32

9

= 26

3 ·

9

32 =

26 · 9

3 · 32 =

13

16

Practice Compute to a single fraction.

1) (a) 3

8 +

7

8 (b)

8

9 –

2

9 (c)

3

8 +

3

5

2) 3

4 –

1

8

3

5 +

2

3 +

3

4

9

16 +

3

8 +

1

4


3) 2 5

8 + 5

7

8 12

1

9 – 7

5

9 6

3

8 + 5

1

4

4) 9 3

20 – 2

4

5

3

4 ·

20

21 4

3

4 · 6

1

3

5) 3

4 ÷

3

5 6

3

4 ÷ 1

1

2

1

2 +

1

3

1

4 +

1

5

Answers-

1) 5/4, 2/3, 39/40

2) 5/8, 121/60, 19/16

3) 8 1/2, 4 5/9, 11 5/8

4) 6 7/20, 5/7, 30 1/12

5) 1 1/4, 4 1/2, 1 23/27

Applied Fractions Practice 1) Find a, b & c

6 1/2"

2 3/4"

Not to Scale 1 7/8"

1 1/2"

2 3/8"

1 1/8"

a b

c

2) Find a,b,c

2"

Ø 516"

1516"

214"

a

b

3"

c

3) Find the total thickness of the floor consisting

of ⅞" sub-floor, ¾" planking to accommodate

in-floor heating, ¼" concrete hardi-plank, ⅜"

porcelain tile with a ⅛" grout line.

1/8" grout line

7/8" comply subfloor

3/4" planking

1/4" concrete board

3/8" tile

4) A ¾ cup of blue and ⅔ cup of green is added to white to make 1 qt (4 cups) of teal paint. How much

white is needed?

5) A 10' pipe (120") is cut into multiple 16 ⅜ inch sections. How many full pieces can be made? What

size piece is left over? Assume 0.0" allowance for the saw cuts.

6) Two 18 ⅞" pieces are cut from 48" stock. Assuming a 1/16" allowance for each saw cut (kerf) what

size piece is left?

Answers-

1) a=3/8, b=2 ¼, 1 ⅝

2) a=3/8, b=3/8, c=7/32

3) 2 ⅜

4) 2 7/12 cup

5) 7 full pieces, 5 ⅜" left

6) 10 ⅛"

1.3 Exercises - Drill with Fractions

Simplify to an exact integer or a reduced fraction. NO DECIMALS!

1) 1

3 +

2

3 2)

8

5 –

2

5 3)

2

5 +

6

5 4) 4 –

2

3

5) 2 ⅝ + ¾ – 1 ⅜ 6) 2

3 –

3

8 +

2

5 7)

8 – 3

40 – 5 8)

6 – 2·3

10 – 2·5

9) 12 ¾

4 10) 3 ⅝ − 2 ¾ 11) 5 ⅞ + 9 ⅔ 12) 3·(1/3 – 3/2) + 4


13)

4

5

15

24

14) 4

3 ·

9

8 15) -6

3

8 · 4

2

3 16)

9

8 ÷

3

2

17) -6 ⅜ ÷ 4 ⅔ 18) 4 ⅞ + 8 ¾ ÷ 2 ½ 19) (12 – 6)·(5 – 4)

(10 + 3)·(12 – 3) 20)

1

2 +

3

4 –

5

4

6 [ 3

2 –

3

4 ]

Applying Fraction Arithmetic

21) Find a

418"

1 716" a

22) Find a

aa

138"138" 2 5

16"2 516"

23) Find a if two equal segments are removed

120"

2258"

a

24) Find "a" assuming all segments are equal

120"

a 25) Find "a" assuming 5 equal segments are removed

140"

934"

a

26) Find "a" assuming 4 equal segments are removed

3 516"

7"

a

27) Find a & b a

b1516"

11116"

234"

212"

28) Find a & b

118"1 7

16"

1 916"

78"

a

b

29) Find a & b

34" 11

2"

dia 3/8"

516"

114"

12"

12"

a

b

30) Find a & b

a

b

158"

1 116"

2 316"

234"

R14"

31) Find a & b

2 916"

2 716"

1 916"

b

118" a

32) Find a & b

33) Find a, b, c, d, r

4.0 "

1.0

"

0.8

"

1.5 "

2.0 "

0.6 "

2.3 "

1.2

"

a

bc

d

r

3/16"

3/16"

34) Find a, b B = 4½", H = 2¼", r = ⅜"

a a a a

B

H

b

b

r


35) Find a, b, c, d

Assume equal spacing of vertical studs in sections

with b & d.

4 ' 2 "

9 ' 6 "

3 ' 4 "

4 ' 0 " a b

8 ' 0 "4 ' 2

"

1 ' 2 "

c

d

36) Find r, a

37) Find the length of the plate in fractional form to the nearest 1/16th inch.

0.625" 0.8125"

1.125"

Ø38" Ø3

8"

38) A 120" pipe is cut into 14" pieces. Each saw cut (kerf) removes ⅛". How many full-size pieces can be

made? How much is left?

39) 78 ⅜" are cut from a 96" piece. Assuming 1

16 " for the saw's kerf, how much is left?

40) From a 96" pipe, two pieces are cut. 19 ⅝" & 12 ¾". Assuming 0.0" allowance for the saw cuts, how much

is left over?

41) A 3 ⅜" motor mount is to be bolted to a frame. If the final thickness must be 4 ¼" what size shim is

needed?

42) A 6 ¾" I-beam is attached to a 2 ½" flange using a ⅛" shim. What is the total thickness?

43) A floor consists of 9¼" TJI's (joists) with a ⅞" comply sub-floor and a ¾" hardwood overlay. What is the

total thickness?

44) A floor is made from 2 ¼" gypcrete with 3/8" ceramic tile on top. Assuming 1/16" for the tile adhesive what

is the thickness of the combined layers?

45) Jeremy is making cookies for the squad. The original recipe calls for 1 ⅓ cups flour. Jeremy wants to

make 5 times the normal batch of cookies. How much flour should he use?

46) Sam wants to mix up 4 qts of epoxy. Each quart needs 1 ⅜ oz of graphite powder. How much graphite

should he add?

47) A mechanism rotates a cog 1 ¾ turn with each activation. How many turns will be made after 514

activations?

48) Three parcels are being logged: 55 ¼ ac, 46 ½ ac and 39 ⅝ ac. What is the total acreage?

49) 27 bricks (8 ¼" long) are laid end to end with a 3/8" grout joint between each pair of bricks. What is the

total length? Note: There is no grout on the ends.

50) A 2 ¼" motor mount is attached to a 1 ⅜" steel plate. Seven ⅜" shims are also used. What is the total

thickness?

51) A 1¾ cup of blue, 3 ⅔ cup of green and ⅛ cup yellow is added to white to make a total of 1 gal (16 cups) of

paint. How much white is needed?

52) Light passes through three filters. The first filter reduces the intensity by ⅜, the second reduces the light by

¼ and the third reduces the light by ⅓. What fraction of light will pass through the triple filter?


1.4 Some Calculator Tips (TI-83 Family)

Making the Screen Lighter or Darker

When you put in fresh batteries, your calculator screen will probably be very dark. As the batteries

wear out, the screen will fade. Adjust it to a comfortable level by pressing 2nd + ▲ to make the screen

darker or 2nd + ▼ to make it lighter. When the number in the upper right corner reaches 9 it is time to

replace your batteries.

Arithmetic

Most (if not all) calculators have a separate subtraction key and a separate negative key. They are not

interchangeable. For instance, on the TI-83+ the subtraction key is blue and is grouped with the other

operators: ^ , ÷ , × , − , + . The negative key is white (-) and is located to the left of the

ENTER key. Type the expression, 'ten minus negative three', (10 − -3) and notice how the subtraction

sign and the negative sign look different on the calculator screen.

Order of Operations

The TI-83+ graphing calculator obeys order of operations, so it will do parentheses first, then

exponents, then multiply and divide and then add and subtract. Press ENTER when you've finished

entering an expression and want an answer.

Parentheses

Only the rounded parentheses ( , ) are used for arithmetic on the calculator. The hard brackets [ ]

and curly brackets { } are used by the calculator for something else. Thus, [(2 + 3) 4 + 5] 6 must be

entered as ((2 + 3) 4 + 5) 6.

Exponents

For exponents, use the x2 key only for squares. Use the ^ key for general exponents. For example,

if you want to compute 32 (which is 9) you would type 3 x

2 ENTER; you'll get 9. If you want to

compute 23 (which is 8) type 2 ^ 3 ENTER; you'll get 8.

Square Roots

Use the 2nd

key in conjunction with the x2 key to get to the square root (written in yellow above the

x2 key). The TI-83+ automatically includes an open parentheses so be sure to close that parentheses at

the appropriate place. For example, if you want to compute 16 + 9 (which is 13) you would type 2nd

x2 16) + 9 ENTER; you'll get 13. If you type 2

nd x

2 16 + 9 ENTER; you'll get 5. Why? Press the

MATH key and choose 4: 3 ( to get a cube root.

Editing

Use the cursor keys, INS and DEL to edit a calculation. Using CLEAR once erases an entire line. Using

CLEAR twice erases the entire screen.

Press 2nd ENTRY (above the ENTER key) to get the calculator to retype the same thing you just typed.

Press it again (and again and again) to get to previous expressions. You can edit and then recalculate

these expressions. If you end up doing something you don't want to do, press 2nd QUIT (above the

MODE key).

Press 2nd ANS (above the negative sign: (-) ) to get the value of the previous answer.


MODE

MODE is used to set the calculator's default display. When you first press

MODE, you'll likely see the screen shown here. All we will discuss here is

Float vs. 0123456789.

Selecting Float causes the display to use the entire screen to most accurately

display a value. e.g. 1/3 → .3333333333 and 1/8 → .125.

Selecting a digit will cause the display to round the displayed value to that decimal place. For example,

0123456789 will cause all results to be displayed rounded to the hundredths place. e.g. 1/3 → .33 and

1/8 → .13

It is important to note that the displayed value is only rounded in the display, its actual stored value is

still as accurate as the calculator can make it.

Storing Results / Retaining Accuracy

The following examples assume we've set the MODE to display our results with two decimals.

0123456789

Suppose we want to compute a complex calculation. The preferred approach is to type the calculation

all in one line or break it into parts. When we break it into part it's common to perform part of the

calculation, write that number on a scratch piece of paper and then retype it as necessary. That approach

generates unnecessary error.

Consider 2 + ⅜

1 − ⅔ which equals 7 ⅛ = 7.125. If we first type 2+3/8 we get 2.38

displayed. If we then type 1−2/3 we get .33 displayed. If we then type

2.38/.33 we get 7.21 which is clearly incorrect. Why?

A better scheme:

Type 2+3/8 STO ALPHA A . Then type 1−2/3 STO ALPHA B . Then type

ALPHA A / ALPHA A and we get 7.13 displayed. Actually, the calculator has

stored 7.125 but the display is rounded to two decimal places.

Storing intermediate steps in the letters A−Z and then recalling them will keep your answers accurate.

When we start a calculation with an operation (such as * ) the calculator

assumes we are continuing the calculation and begins the calculation with the

last answer (Ans) and goes forward. No accuracy is lost. Here we compute

3(32−4×5) by breaking the calculation into parts.

Fractions

To enter a non-mixed number, use the divide button. e.g ⅔ is entered as 2 ÷ 3 ENTER. To enter a

mixed number, use addition to connect the whole and the fractional parts. e.g. 1½ is entered as 1+1÷2

ENTER. Why?

Most modern calculators (e.g. TI-83+) have built-in fraction features which allow one to use a calculator

and maintain fractions format. On the TI-83+ use the MATH button and then the [MATH] category to


select Frac. This enters the Frac command on the screen. Hitting ENTER executes that command

and displays the current screen value as an appropriate improper fraction.

For example, suppose .375 is currently displayed on the screen. Selecting Frac will generate

AnsFrac. ENTER yields 3/10.

Frac may also be used as part of a computation in progress. For example, 3/10Frac ENTER → 3/10

rather than .3

Example 1 Simplify using the TI-83+

1) 3

5 +

2

3 +

-3

4

2)

5

8 –

2

3

3) 6 3

8 + 5

1

4

4) 9 3

20 – 2

4

5

5) 3

4 ·

20

21

6) 4

3

4 · -6

1

3

7) 3

4 ÷

-3

5

8) 6

3

4 ÷ 1

1

2

9)

1

2 +

1

3

1

4 ×

1

5

10) 387/6 = 55 2

7

Calculators encourage their own kinds of mistakes. Listed below are some of the places where mistakes are

commonly made.

Beware!

When entering calculations with mixed numbers it is often necessary to use parentheses to

ensure correct order of operations. For example:

1½ × 2¾ is entered as (1+1/2)(2+3/4). -1½ × -2¾ is entered as -(1+1/2)-(2+3/4). Both "-"

symbols are negative keys, not subtraction! -1½ is entered as -(1+1/2). The negative must be

outside the parentheses. Why? 1½ − 2¾ is entered as (1+1/2)−(2+3/4). What happens if the

parentheses are omitted?

To avoid mistakes when using a calculator wrap all mixed fraction in a parentheses.

-1⅔ CANNOT be entered into a calculator as -1+2/3. Why?

-1⅔ → -(1+2/3)

8 − 5 ⅔ CANNOT be entered into a calculator as 8−5+2/3. Why?


8 − 5 ⅔ → 8−(5+2/3)

7 ÷ 1½ CANNOT be entered into a calculator as 7/1+1/2. Why?

7 ÷ 1½ → 7/(1+1/2)

4(-1⅔) CANNOT be entered into a calculator as 4*-1+2/3. Why?

4(-1⅔) → 4*-(1+2/3)

Watch out for other grouping symbols besides

parentheses. Fraction dividers and radicals imply a

grouping and imply parentheses. Since we so often

use a calculator to key in such expressions to

preserve order of operations (PEMDAS) get in the

habit of including the necessary parentheses when

using a calculator.

expression

expression → (expression)/(expression)

expression → (expression)

6 · 15

25 · 9 CANNOT be entered into a calculator as 6*15/25*9. Why?

6 · 15

25 · 9 → (6 * 15) / (25 * 9)

4 + 10

10 − 3 CANNOT be entered into a calculator as 4+10/10−3. Why?

4 + 10

10 − 3 → (4+10)/(10−3)

3(32−4×5) CANNOT be entered into a calculator as 3(32−4×5). Why?

3(32−4×5) → (3(32−4×5))

When in doubt use extra parentheses to remove any chance of ambiguity and to help the reader

avoid mistakes. But when using a calculator only rounded parentheses ( ) may be used for

arithmetic in a calculator. DO NOT use { } or [ ] for calculator arithmetic.

Most (if not all) calculators have an exponentiation key. On the TI-83+ the exponentiation key

is black: ^ . To compute (-2)(-2)(-2)(-2) = (-2)4 use (-2)^4 = 16. If you mistakenly omit the

parentheses and type -24 the TI-83+ computes the result as -16. Why? Notice that the

parentheses were crucial.

What is the difference between (-a)n vs. -a

n ? Notice that parentheses make a difference.

(-3)2 = (-3)(-3) = 9 these are different! -3

2 = -(3 · 3) = -9

-32 ≠ (-3)(-3) this is a very common mistake!

Remember, extra parentheses will not cause an error. Leaving them out often does.


Example 2 Translate into Calculator Format '-' negative key, '−' subtraction key

Printed Version → Entered into TI-83+ Printed Version → Entered into TI-83+

4[3 + 5(2 + 1)] → 4(3 + 5(2 + 1)) {2 + 3[1 + (2)(3)]} → (2+3(1+(2)(3)))

4

2 + 5 → 4/(2 + 5)

2 + 3

3 + 4 → (2+3)/(3+4)

4

5π → 4/5/π or 4/(5π)

1

1½ → 1/(1+1/2) or 1/1.5

9 + 16 → (9+16) 1⅔ × 3¾ → (1+2/3)(3+3/4)

-1⅔ + -3¾ → -(1+2/3)+-(3+3/4) 1⅔

3¾ → (1+2/3)/(3+3/4)

1⅔ − 3¾ → (1+2/3)−(3+3/4) 10

10 → (10)/10

1.4 Exercises - Calculator Arithmetic

Use a calculator to simplify to a decimal number accurate to the hundredths place.

1) 7 + 8

8 − -12 2)

27

9 − 3 · 4 3)

37 + (3)(7)

(2)(-3)(4) − 5

4) 4 + 8

8 − 4 5)

4 + 8

8 − 2 · 4 6)

3 · 4 − 6 · 2

8 − 4

7) 5 + 5[50 − 5(10 − 5)] 8) 2+2{2+2[2+2(2+2)]} 9) 1 +

1

1 + 1

1 + 1

10) -2(-3) + (-2)

3 + (-3)(5) 11)

-2(-4) + (-3)3

5 + (-3)(2) 12)

-2(4) + (-3)

-7 + (-3)(5)

13) 3(4) – 12

2 – 2 14)

(-3) + 2(-6)

3 + 8(-6) 15)

3(-2) + 6

37(-2)

16) (-4)3 17) [-3(5–7)+(-2)]6÷16–15 18)

1 + ¾

2 − ⅔

19) 1

2 2 20) [-3(5 – 8)+4]4÷13−26 21)

2

3 5

22) 1

2π 23)

3

4π 24)

23 – 32

4·6 – 52

25) (-10)5 26) (-0.1)

6 27) (¼ − ¾)

4

28) 2(-3) + (-10)

3 + (-3)(-5) 29)

-2(-4) + (-3)3

5 + (-3)2 30)

-24 + (-3)

-7 + (-3)2

31) 3(4) – (-2)(-6)

9 – 2 32)

(-7) + (-2)(-6)

3 + 8(-6) 33)

3(-2) + 5 + 18

37(-2)

34) (32 + 4)

2 35) (2

3 − 3

2)

3 36) (-1)

27

37) -2.3 + 1.2

5.4 – 2.7 38)

-1.5 + 5.2

7.32 · 12.17 39)

-1.1 − 4.22

-6.2 + 1.4 + 5.54

40) π 10 41) -π

2

6.23 + π2 42) -π

4 + 2400 π – 3

43) 2.5 + 6.5

4.8 – 1.2 44) (3.69 – 2.61)9.61 – 1.44 45) (1.35 – 7.23)(4.44 – 9.22)

46) 6 + 4

12 – 3 47) (6.25 – 4.36)(9.24 – 3.54) 48)

8.4 + 9.2

20.6 – 4.2


49) 1.52 + 4.27

3.11 – 5.73 50) 6 + 4 12 + 3 4 + 5 51)

- -4.25 + 9.24

6.25 – 3.50

52) 4.25 + 9.24

6.25 – 2.50 53) 2.5+6.5 -4.8+4.2 5.3+9.6 54) (2.5 + 6.52 + 8.4)

1.5

55) (15 + 42 + 25)3/2

56) -8.4+9.2 2.6+4.2 4.3+8.2 57) (-8.44 + 9.272 + 5.45)3/2

58) 52 + 11

2(6 – 3) 59)

12.52 + 78.4

2(9.2 – 7.3) 60)

25 + 43

2 (6.2 – 1.5)

61) 132 – 122 62) -5[-4(6 + 2) – 7]8 – 8 63) 14.32 – 12.22

64) 102 – 62 65) 2.2[3.5(4.2+7.5)–4.4]9.2–7.2 66) (23 + 12)2

2 – 32

67) (-6 + 12)·5

17 – 42 68)

30

(2.8)(1.2) 69)

(6.54 + 7.89) 1.23

4.55 – 2.882

70)

8 + 5

9

8 – 5

8

71)

1.4 + 2.5

3.6

7.8 – 8.2

3.5

72)

547 + 986

12

587 – 896

23

73) (-2.34)(1.292)

5.43 – 2.17 + 6.39 74)

(-1.45)(5.282)

7.32 – 12.17 – 7.39 75)

(-1.11)(4.292)

-6.22 + 1.44 + 5.54

76) 2 2 2 77) 3 3 3 78) 1+2 1+2 1+2

79) ½ ⅓ ¼ 80) 8(13 − 5) 81) 2(3 + 5)

82) 35 − 10 83) 8 + 8 84) 2 8 + 9

85) (½)·(⅔)·(¾) 86) (-1)8 87) 8 ÷ 10

88) 8 + 8 89) 30 + 29 90) (½) 24

Use your calculator to simplify to a single fraction. Write improper fractions as mixed fractions.

91) 1

3 +

2

3 92)

8

5 –

2

5 93)

2

5 +

6

5

94) 4 – 2

3 95) 2 ⅝ + ¾ – 1 ⅜ 96)

2

3 –

3

8 +

2

5

97) 8 – 3

40 – 5 98)

6 – 2·3

10 – 2·5 99)

12 ¾

4

100) 3 ⅝ − 2 ¾ 101) 5 ⅞ + 9 ⅔ 102) 3·(1/3 – 3/2) + 4

103) 4/5

15/24 104)

4

3 ·

9

8 105) -6

3

8 · 4

2

3

106) 9

8 ÷

3

2 107) -6 ⅜ ÷ 4 ⅔ 108) 4 ⅞ + 8 ¾ ÷ 2 ½

109) (12 – 6)·(5 – 4)

(10 + 3)·(12 – 3) 110)

1

2 +

3

4 –

5

4

6 [ 3

2 –

3

4 ]

111) 8 + ¾

8 – ⅝

112)

4 + 5 · 3

10

7 – 2 · 8

5

113)

47 − 45 × 9

2

57 – 55 × 8

3

114)

4½

2⅓

6¾

2⅝


Applied Order of Operations Exercises- Write a valid expression for each statement then compute the result

115) A 22' pipe has a 7'and 5'piece cut off. How

much remains?

116) A welder must weld three lengths together; 42"

37" and 16". How long is the resulting piece?

117) A hybrid car that gets 52 mpg (miles per gallon)

has a 12 gal gas tank. How many miles could

the car travel on one tank full of gas?

118) A V8 has a total displacement of 2000 cu-in.

What is the displacement of each cylinder?

119) The following amounts of hydraulic fluid have

been recorded. Inputs: 32 cc, 24 cc, 92 cc, 18cc

Outputs: 16cc, 27cc, 104cc, 19 cc. What is the

net result?

120) A farmer harvests 232 bushels per acre on

average for his 240 acre parcel. He sells the

wheat for $2.38 per bushel. What is his return?

121) Kermin orders 1,256' of siding. If each piece is

12' long, how many pieces should be delivered?

122) Sydne has 3 regular employees earning

$8.65/hr and a supervisor earning $13.25/hr.

What is the total payroll for a 40 hr week?

123) A plane can climb at 420 ft/min. How long will

it take the plane to gain 5200 ft?

124) A plane can climb at 420 ft/min. How far will

it climb in 15 min?

125) Paul has four boxes of washers with the

following quantities: 143, 162, 154, 137. What

is the average number of washers per box?

126) What is the formula for the average of the four

amounts; a, b, c, d?

127) A 17' wall is to have three 3' windows evenly

spaced (including the ends). How much space

is between each window?

128) A 23' wall is to have two picture windows

spaced 4 feet apart. What is the spacing on

each end of the wall if the windows are 5'

wide?

129) A nurse administers 6 gm/hr of a sedative. How

much will be administered in 4 hrs?

130) A nurse administers 12 gm/hr of a drug. How

long until the patients has received 200 gms?


1.5 Squares, Cubes & Roots and Exponential Notation

The square of a number “a” is (a)(a). e.g. the square of 7 is (7)(7) or 49. The cube of a number “b” is

(b)(b)(b). e.g. the cube of 5 is (5)(5)(5) or 125. These terms arise from the geometric relationships for a

square and a cube. A square with dimensions a × a has an area A = a·a or a-squared. Similarly, a cube

with dimensions b × b × b has a volume V = (b)(b)(b) or b-cubed.

The regular occurrence of geometric squares and cubes all around us should give us some indication of

how frequently we will encounter the algebraic expressions of squares and cubes. You should not be too

surprised then to learn that there are other symbols and terminology for exactly the same thing.

1.5.1 Exponential Notation

Repeated multiplication (of one value) occurs quite often in mathematics and a shorthand notation is

commonly used to represent such a product. a·a·a·a·a·…·a = an where n is the number of factors of “a”

present in the product. “a” is called the base and “n” is called the exponent or power. e.g. 6·6·6·6 = 64.

“6” is the base and “4” is the exponent. This shorthand notation is called exponential notation.

There are a variety of phrases and notations we commonly use for exponential notation. All of the

following are equivalent.

5·5·5 = 53 = 5 cubed = 5 to the third power = 5 to the 3

rd = 5 to the 3 = 5^3 = 125

Note: 5^3 is usually only used where it’s inconvenient to write “53“ such as electronic displays.

1.5.2 Roots & Radicals

The second root or square root (or simply root) of a number is the positive value which when

multiplied by itself returns that number. For instance, the square root of 9 is +3 because 3·3 returns 9.

Although there are always two potential answers for a square root, the negative root is considered

extraneous while the positive root is considered the primary root. The squared root is explicitly defined

to be the positive choice. If we want the negative root we should explicitly say the negative root of…

For instance, the root of 25 is +5. However, the "negative root of 25" is -5 because (-5)(-5) = 25.

In a similar fashion, the third root (or cubed root) of a number is the value which when multiplied by

itself three times returns that number. For instance, the third root of 64 is 4 because 4·4·4 yields 64 and

the third root of -8 is -2 because (-2)(-2)(-2) = -8.

We use this same definition for other positive whole number roots. For instance, the 9th

root of

40,353,607 is “7” because 7·7·7·7·7·7·7·7·7 = 79 = 40,353,607. Because the second and third roots are

quite common they have equivalent names square root and cube root respectively. The term radical

refers to any expression involving a root.

The common notation used for roots is root

radicand as in 3

125 = 5. Since the square root is the most

common we often omit the root as in 64 = 8.

Unfortunately, not many roots may be expressed as an exact fraction or an exact decimal. For instance,

1.41 < 2 < 1.42. And even 1.41421356237 is only a very close approximation to 2 . This was very

disconcerting to some early mathematicians as it might be to some of you now. For this reason we refer


to all numbers which cannot be represented as an exact fraction or as a decimal as irrational numbers.

Conversely, all numbers that may be represented as an exact fraction or decimal are called rational

numbers. We shall examine irrational numbers as well as additional notations and properties of

exponents and roots in greater detail in a later section.

1.5.3 Order of Operations with Exponents and Radicals

Roots and exponents share the same priority and are superseded only by parentheses. One must be

careful to note where the root symbol ends as the root symbol acts as a grouping and therefore implies

parentheses. To be on the safe side, add parentheses to your radical expressions and DO NOT write

anything after a root symbol if you can help it. If you must write an expression after a root symbol use a

parentheses to clarify the order of operations.

Example 1 7 · 5 could be mistaken for 7·5 which is incorrect. So write 5· 7 or ( 7 )·5.

Example 2 16 + 9 could be mistaken for 16 + 9 which is incorrect. So write 9 + 16 or ( 16 )·9

1.5.4 Roots and Radicals with Fractions

For simple fractions we can compute the square, cube or a root by hand. [a

b ]2

= [a

b ][a

b ] = a2

b2 Similarly,

we can compute the roots by hand. a

b =

a

b .

Example 3 [ 1

2 ]

2

= 12

22 = 1

4 [ 3

8 ]2

= 32

82 = 9

64 9

16 = 9

16 =

3

4

Beware! Expressions of Roots and Powers cannot be simplified by a shortcut when addition or subtraction occurs

within the expression. These are common mistakes. Don't get caught making them.

(a + b)2 ≠ a

2 + b

2 (a + b)

3 ≠ a

3 + b

3 (10 + x)

2 ≠ 100 + x

2

a2 + b

2 ≠ a + b a + b ≠ a + b a − b ≠ a − b

Practice 1) 3

5 = 6^3 = 4 to the 5

th = 10 squared =

2) 42 · 2

4 = -5

2 = 8

2 ÷ 4 = 10

2 − 8

2 =

3) 100 − 52 = 5 cubed = 0.5

2 = [

2

5 ]

2

=

4) 0.1252 = [

2

52 ] = 10

6 = 0.01

3 =

5) What is the common name of 104



8) Write the algebraic expression (in exponential notation) for the area of this

square. A = . a

9) Write the algebraic expression (in exponential notation) for the volume of this

cube. V = a


10) Write the expression in exponential notation for 8·8·8·8·8 =

11) Write the expression in exponential notation for 2·2·3·3·3 =

12) 169 = 4

625 = Approximate 20 to the nearest thousandth

13) 16 + 9 = 144 + 25 = 32

1 + 2·3 + 32 4 =

14) 132 − 5

2 = 24

2 + 7

2 = 3

2 + 4

2 + 12

2 =

Answers-

1) 243, 216, 1,024, 100

2) 256, -25, 16, 36

3) 75, 125, 0.25, 4/25 or 0.16

4) 0.015625, 2/25,

1,000,000, 0.000001

5) ten thousand

6) one million

7) one billion

8) a2

9) a3

10) 85

11) 22·3

3

12) 13, 5, 4.472

13) 5, 37, 45

14) 12, 25, 13

Perhaps it is already obvious that roots and exponents are closely related. They are inverse operations.

Multiplication and division are another example of inverse operations. Can you think of any others?

We will discuss inverse operations in greater detail in a later chapter.

1.5 Exercises - Squares, Cubes and Radicals

Simplify to a single number without exponents or radicals. 1) a) 2

5 = b) 3^3 = c) 4 to the 3

rd = d) 10 cubed =

2) 32 · 2

3 = 10 − 4

2 = 10

2 ÷ 5 = 11

2 − 8

2 =

3) 100 − 102 =

42

32 = 0.25

2 = [

2

3 ]

2

=

4) 0.12 = [

2

52 ] = 10

3 = 0.01

2 =



7) Write the algebraic expression (in exponential notation) for the area of this square.

A = w 8) Write the algebraic expression (in exponential notation) for the volume of this cube.

V = w

9) Write the expression in exponential notation: 8·8·8·8·8·8·8 =

10) Write the expression in exponential notation for 2·2·2·3·3·3·3·3 =

11) 144 = 4

16 = 64

25 =

12) 125 − 4 = 64 + 25 = 2·3 +1 + 3 25 − 6 =

13) 252 − 24

2 = Approximate 2 to the nearest thousandth: 2 ≈

14) 8 32 = [ 1 ]10

= 34 =

15) Approximate 2

2 to the nearest thousandth:

2

2 ≈

16) Approximate ½ to the nearest thousandth: ½ ≈


1.6 Powers of 10 and Scientific Notation

Considering our numbering system (decimal) is based on 10 it is not surprising that one of the most

common uses of exponents is with powers of 10. 10n = 10·10·10·…·10 (n-times). Within exponential

forms, powers-of-ten are somewhat special. There are common names for powers of 10 and you should

know them.

Common Powers of 10

1012

= 1,000,000,000,000 one trillion

109 = 1,000,000,000 one billion

106 = 1,000,000 one million

105 = 100,000 one hundred thousand

104 = 10,000 ten thousand

103 = 1,000 one thousand

102 = 100 one hundred

101 = 10 ten

? = 1 one

? = 0.1 one tenth

? = 0.01 one hundredth

? = 0.001 one thousandth

How should the pattern be continued? The only reasonable choices are 100 = 1, 10

-1 = 0.1, 10

-2 = 0.01,

etc. This leads to a natural extension of exponential notation to negative integers. Namely, negative

exponents are interpreted as repeated division.

Common Powers of 10

103 = 1,000. from 1.0 decimal point shifted +3 places

102 = 100. from 1.0 decimal point shifted +2 places

101 = 10. from 1.0 decimal point shifted +1 places

100 = 1. from 1.0 decimal point shifted 0 places

10-1

= 0.1 = 1

10 =

1

101 from 1.0 decimal point shifted -1 places

10-2

= 0.01 = 1

100 =

1


10-3

= 0.001 = 1

1000 =

1


Positive exponents indicate repeated multiplication while negative exponents indicate repeated division with 10

0 =

1. In general:

an = a · a · a · a · …· a (n-times) a

0 = 1 (a ≠ 0) a

-n =

1

an =

1

a · a · a · a ·…· a (n-times)


Example 1

(a) 35 = 3 × 3 × 3 × 3 × 3 = 243 (b) 8

-3 =

1

83 =

1

8 × 8 × 8 =

1

512 (c) 7

0 = 1

(d) -50 = -1 (e) 27 = 3 × 3 × 3 = 3

3 (f) -

1

2 =

-1

2 × 2 × 2 =

-1

23 = -2

-3

Shortcuts with Powers of 10

We can quickly write the decimal form for a power-of-ten by taking advantage of the shortcut for

multiplication or division by 10. Each multiplication by 10 shifts the decimal one place to the right

while each division by 10 shifts the decimal point one place to the left. Thus, the exponent simply

represents the decimal shift. Note: This shortcut is true only for a base of 10.

Beware!

You might be tempted to associate the exponent with the number of zeros that occur when 10n is

written in decimal form. That can lead to mistakes especially when 10n appears in a calculation.

Note 5.2 × 10-3

= .0052 has only two zeros and 3.4 × 103 = 3400. has only two zeros. However,

in each case, the decimal point is shifted three places to correspond with the exponent.

The sign of 10n is independent of the exponent. e.g. -10

2 = -100 and -10

-2 = -1/100. Do not

confuse the sign of the exponent with the sign of the number.

Example 2

(a) 105 = 100,000 (b) 3.8 × 10

3 = 3,800 (c) -10

-4 = -0.0001

(d) 18,000 = 1.8 × 104 (e) -1.2 = -1.2 × 10

0 (f) 0.00071 = 7.1 × 10

-4

Practice

1) Convert to decimal format:

(a) 103 (b) 10

-4 (c) 10

-6 (d) -10

1

2) Convert to exponential format:

(a) 10,000 (b) 0.00001 (c) 0.000001 (d) -1000

3) Give the common name for these powers of 10:

(a) 10-3

(b) 109 (c) 10

-6 (d) -10

1

Answers-

1) 1,000, 0.0001, 0.000001, -0.1 2) 104, 10-5, 10-6, -103 3) one thousandth, one billion, one millionth, negative ten

1.6.1 General Exponents

Now that we have a basic understanding of how the abbreviation of exponential notation (sometimes

called power notation) naturally extends to integer powers we are ready to expand this to a multitude of

relationships and shortcuts for calculations involving exponential notation.

Terminology: an

baseexponent

x2 = "x squared" x

3 = "x cubed"


Exponential Rules and Shortcuts

n times

1 an = (a · a · . . . · a)

n times m times

m+n times

2 a

n · a

m = a

n + m (a · a · . . . · a)(a · a · . . . · a) = (a · a · . . . · a · a · a · . . . · a)

3 an/a

m = a

n–m

a · a · a . . . · a

a · a =

a · a · a . . . · a

a · a = a . . . · a (n–m a's after canceling)

4 a0 = 1 when a ≠ 0 by definition, 0

0 is undefined

5 a–n

= 1

an ,

1

a–n = a

n negative exponents indicate repeated division!

6 (a · b)n = a

n · b

n each factor is raised to the nth power

7 ( a

b )

n

= a

n

bn each factor is raised to the nth power

8 (ab)c

= ab·c Note: (a ± b)

n ≠ a

n ± b

n

9 (xa · y

b)c = x

ac · y

bc rule 6 & 8 combined

10 a = a0.5

fractional exponents indicate roots!

11 na = a

1/n fractional exponents indicate roots!

12 na

m = a

m/n fractional exponents indicate roots!

Memorizing these rules is counter-productive if they do not seem logical. You will become adept with

exponential notation once these rules seem logical so focus on understanding the shortcuts, rather than

memorizing them. Remember, we can always write out every exponent using definition 1 and count the

resulting factors. Doing so should eventually make the shortcuts appear obvious.

Beware! Watch out for these common mistakes-

(a ± b)n ≠ a

n ± b

n a ± b ≠ a ± b a

0 ≠ 0

(x + 5)2 ≠ x

2 + 25 3

2 + 4

2 ≠ 3 + 4 = 7 10

0 ≠ 0

Example 3

23 · 2

5 = 2

3+5 = 2

8 x

2 · x

3 = x

2+3 = x

5 4

8 · (¼)

8 = (4 · ¼)

8 = (1)

8 = 1

2-3

= 1

23 =

1

8 = 0.125 10

2 · 10

-3 =

102

103 =

1

10 = 10

-1

x2 · y

4

x5 · y

2 = y

2

x3

(2x3)

2 = 2

2 (x

3)

2 = 4x

6 (5·10

3)

2 = 5

2 · (10

3)

2 = 25·10

6 4x

3 = 4

½ x

1.5 = 2 x

1.5

102 · 10

-6 = 10

2 + (-6) = 10

-4 =

1

104

9 · 106 = (9 · 10

6)

½ =

9½ · (10

6)

½ = 3 · 10

3

10-3

= 1

103


Practice- Simplify (removing parentheses and roots) and convert to positive exponents

1) 85 · 8

2 =

2) x2 · x

7 =

3) (8x)4 · (⅜)

4 =

4) -5-2

=

5) 107 · 10

-5 =

6) (17,384,678)0 =

7) (5x3)

2 =

8) (7·103)

2 =

9) 100x5 =

10) 10

12

1016 =

11) -z

-5

z3 =

12) x

5 · y

4

x4 · y

7 =

Answers-

1) 87

2) x9

3) 34 x

4

4) -1

52

5) 102

6) 1

7) 52 x

6

8) 72 10

6

9) 10 x2.5

10) 1

104

11) -1

z8

12) x

y3

×

1.6.2 Scientific Notation

The inconvenience of writing very large (or small) numbers combined with our penchant for shorthand

has led to an abbreviation known as scientific notation. Since multiplication (or division) by powers of

10 can be quickly done with a simple shift of the decimal point very large (or small) numbers can be

written quite compactly. For instance, 1.5 trillion = 1,500,000,000,000 = 1.5 × 1012

. This exponential

format is called scientific notation. When we multiply (or divide) by a power of 10 we shift the decimal

once for each factor of 10. Thus the decimal shift aligns exactly with the exponent; positive exponents

indicate a shift to the right and negative exponents indicate a shift to the left.

± X . X X X X · 10 ± n

sign of number

non-zero leading digit

significant digits

integer exponent

direction for decimal point shift

Example 4

2.345 × 103 = 2345. 10

3 indicates we shift the decimal +3 places

2.345 × 102 = 234.5 10

2 indicates we shift the decimal +2 places

2.345 × 101 = 23.45 10

1 indicates we shift the decimal +1 place

2.345 × 100 = 2.345 10

0 indicates we shift the decimal 0 places

2.345 × 10–1

= 0.2345 10–1

indicates we shift the decimal -1 places

2.345 × 10–2

= 0.02345 10-2


2.345 × 10–3

= 0.002345 10-3



Practice Write an equivalent value in decimal format

1) 3.65 × 103 = 2) 3.57 × 10

-4 = 3) -6.89 × 10

-1 =

4) (2 · 103)(5 · 10

-4) 5) (3 × 10

-3)

2 = 6) 3 × 10

4 + 5 × 10

3 =

Write an equivalent value in Scientific Notation

7) 37,800 = 8) 6.7 milliamps = 9) 5 megawatts =

10) 2.4 billion = 11) 0.0398 = 12) -0.003 =

Answers- 1) 3,650

2) 0.000357

3) -0.689

4) 1

5) 0.000009

6) 35,000

7) 3.78 × 104

8) 6.7 × 10-3 amps

9) 5 × 106 watts

10) 2.4 × 109

11) 3.98 × 10-2

12) -3 × 10-3

1.6.3 Arithmetic with Scientific Notation

Although we expect to use a calculator for complex calculations we should be able to compute simple

calculations involving scientific notation by hand. If we follow order of operations we should be okay

but it might be worth making note of the few shortcuts mentioned here.

Arithmetic Shortcuts

We can change the power of ten and compensate by changing the decimal point and

visa-versa. Increase the power ↔ decrease the decimal or decrease the power ↔

increase the decimal.

When adding (or subtracting) we want to have the same powers of 10. Then we can

simply add (or subtract) the significant digits and then put the result back together.

When multiplying (or dividing), separate the significant digits from the powers of 10.

Now multiply (or divide) each part and then put the result back together.

When raising by a power, separate the significant digits from the powers of 10. Now

raise each part to the power and then put the result back together.

When taking a root, first convert to an even power of 10. Now raise each part to the

half power and then put the result back together.

Example 5

(a) 600 × 105 =

6 × 107

shift decimal -2 places

shift power by +2

(b) 8250 × 10-7

=

8.25 × 10-4

shift decimal -3 places

shift power by +3

(c) 0.002 × 105 =

2 × 102

shift decimal +3 places

shift power by -3

(d) 6.5 × 105 + 5.7 × 10

5 =

(6.5 + 5.7) × 105 =

12.2 × 105 =

1.22 × 106

(e) 3.7 × 104 + 8 × 10

3 =

3.7 × 104 + 0.8 × 10

4 =

(3.7 + 0.8) × 104 =

4.5 × 104

(f) 7.2 × 10-3

− 6 × 10-4

=

7.2 × 10-3

− 0.6 × 10-3

=

(7.2 − 0.6) × 10-3

=

6.6 × 10-3


(g) (4.2 × 105)(2.5 × 10

3) =

(4.2 × 2.5)(105 × 10

3) =

10.5 × 108 =

1.05 × 109

(h) (1.8 × 105)(7.5 × 10

-3) =

(1.8 × 7.5)(105 × 10

-3) =

13.5 × 102 =

1.35 × 103

(i) 7 × (2.5 × 103) =

(7 × 2.5)(103) =

17.5 × 103 =

1.75 × 104

(j) 9.6 × 10

8

1.2 × 102 =

9.6

1.2 ×

108

102 =

8 × 106

(k) (4.2 × 105)

2 =

(4.2)2 × (10

5)

2 =

17.64 × 1010

=

1.764 × 1011

(l) 2.5 × 107 =

25 × 106 =

25 × 106 =

5 × 103

Practice- Answer in Scientific Notation

1) 7200 × 105 = 5) 6.9 × 10

4 + 7 × 10

3 = 9) 15 × (9.6 × 10

-2) =

2) 390 × 10-7

= 6) 7 × 10-5

− 6.4 × 10-4

= 10) 4.7 × 10

5

9.4 × 102 =

3) 0.00045 × 108 = 7) (8.2 × 10

5)(5.5 × 10

2) = 11) (8 × 10

5)

2 =

4) 8.3 × 105 + 5.7 × 10

5 = 8) (2.8 × 10

-7)(7.5 × 10

3) = 12) 8.1 × 10

11 =

Answers-

1) 7.2 · 108

2) 3.9 · 10-5

3) 4.5 · 104

4) 1.4 · 106

5) 7.6 · 104

6) -5.7 · 10-4

7) 4.51 · 108

8) 2.1 · 10-3

9) 1.44 · 100

10) 5 · 102

11) 6.4 · 1011

12) 9 · 105

1.6.4 Scientific Notation on TI-Calculators

Most calculators accommodate scientific notation. On TI graphing calculators use the MODE key to

access the display formats. You can toggle between Scientific Mode and Normal Mode. You can use

2nd ANS to get the TI-83+ to redisplay the answer after changing the format. For the following practice

problems and exercise set Mode to Sci & 0123456789

Normal Mode with

2 decimal places

Typical Normal

Display

Typical Scientific Display Scientific Mode with

2 decimal places

To enter 1.23 · 105 type: 123000 or 1.23 × 10 ^ 5 or 1.23EE5. You should get 1.23E5

On the TI calculators E5 denotes 105. Note that the alpha numeric "E" and the scientific notation "E" are

completely different. They are not interchangeable. Use either 10^ or EE but not both at the same

time. To obtain "E" use 2nd EE. Practice with your calculator on the computations of Example 5 until

you are completely confident you can correctly use scientific notation on your calculator.


Example 6 Using Scientific Notation

1) Estimate the Earth's volume in cubic feet. Use V = 4 π r

3

3 and a radius of ≈ 4,000 mi.

(a) First find the radius by converting 4,000 miles → feet. 4000 mi × 5280 ft

mi = 2.112 × 10

7 ft

(b) V = (⅓) 4 π (2.112 × 107 ft)

3 = (⅓) 4 π (2.112 × 10

7)

3 ft

3 ≈ 4 · 10

22 ft

3

2) Estimate the gravity constant G. Use G = F (RE)

2

ME·m where RE = Earth's radius ≈ 6.37 × 10

6 m,

ME = Earth's mass ≈ 5.98 × 1024

kg, F ≈ 9.807 N and m = 1 kg. i.e. At the Earth's surface there is a

force of ≈ 9.807 N acting on a mass of 1 kg.

G = 9.807 N [ (6.37 × 10

6 m)

2

(5.98 × 1024

kg)( 1 kg) ] =

(9.807)(6.37)2(10

6)

2 N-m

2

(5.98)(1024

)kg2 ≈ 6.7 × 10

-11 N-m

2

kg2

3) Estimate the time it takes for sunlight to reach the Earth. Use T = DES

c where c = the speed of light ≈

3.0 × 108 m/sec. DES = Earth-Sun distance ≈ 1.5 × 10

11 m.

T = 1.5 × 10

11 m

3.0 × 108 m/sec

≈ 5.0 × 102 sec ≈ 8 ⅓ min

4) Estimate the amount of fresh water needed on a yearly basis. Assume 7 billion people on Earth

needing an average of 50 gal/day. Use 365 days/yr.

[7 × 109 people][ 50 gal

person-day][365 days

yr] = (7)(10

9)(50)(365) gal/yr ≈ 1.3 × 10

14 gal/yr

Practice

1) Estimate the amount of fresh water (gallons) contained in Crater Lake. Use V = π R

2 H

3 (the volume

of a cone), where the radius of Crater Lake, R ≈ 1.5 · 104 ft, the depth of Crater Lake, H ≈ 1,932 ft

and ≈ 7.5 gal/cu-ft.

2) Estimate the number of seedlings it would take to reforest the Biscuit fire, SW Oregon, 2002. The fire

burned about one half million acres. Assume one seedling every 10 sq-ft. Fighting the fire cost about

$150 million. What is the additional reforestation cost if each seedling + labor costs $2.50?

3) Each ⅛th inch of a phone book's white pages contains about 100 pages with each page having 3

columns. On average, there are 10" of text per page with 11 phone numbers per inch. Estimate the

number of phone numbers in 2 ¾" of white pages.

Answers-

1) 3 ×1012

gal 2) 2 × 109 seedlings, $5 billion 3) 7 · 10

5 numbers


1.6 Exercises -

Rewrite as a decimal number

1) 1.35 × 105 2) 5.28 × 10

-3 3) -7.42 × 10

-1 4) (5 × 10

3)(7 × 10

-4)

5) (6 × 10–3

)2 6) 6 × 10

5 + 5 × 10

4 7) -4.35 × 10

0 8) (8 × 10

3) ÷ (5 × 10

-4)

Rewrite in Scientific Notation

9) 28,800 10) 7,400,000 11) 20 million 12) 7.4 billion

13) 0.0368 14) -0.0050 15) 74 thousandths 16) 125 hundredths

Compute and give your answer in Scientific Notation

17) 680 × 105 18) 92000 × 10

-4 19) 0.00024 × 10

6 20) 0.0035 × 10

2

21) 3,090 × 10-7

22) 1.8 (9.6 × 10-2

) 23) 7 × 104 + 8 × 10

3 24) 6 × 10

-3 + 8 × 10

-2

25) 8 × 10-5

− 6.4 × 10-4

26) 7 × 105 − 6.4 × 10

4 27) (2.8 × 10

-7)(7.5 × 10

3) 28) (9.2 × 10

5)(8.5 × 10

2)

29) (7.4 × 105)

2 30) (2 × 10

3)

-4 31)

5.7 × 105

9.4 × 102 32)

1.2 × 105

4 × 10-2

33) 8 ×10

5 + 5 × 10

5

(9 ×105)(8 × 10

2) 34)

8 × 10-2

+ 5 × 10-1

(9 × 102)(8 × 10

0) 35)

(4 × 10-8

)(8 × 1011

)

7 × 104 + 8 ×10

3 36) (3 × 10

2)(8 × 10

1)

5 × 102 + 8 × 10

3

37) Estimate the Earth's volume in cc's. Use V = 4 π r

3

3 , a radius of 4,000 miles and 1 mi ≈ 1.609 × 10

5 cm.

38) Estimate the time it takes for a signal from Earth to reach a Moon orbiter. Use T = DEM

c and the speed of

light, c ≈ 3.0 · 108 m/sec, the Earth Moon distance, DEM ≈ 1.8 × 10

5 miles and 1 mi ≈ 1.609 × 10

3 m.

39) Estimate the amount of gasoline used per year. Use 7 billion people with ⅛th of them driving. Assume

each driver averages 40 mi/day with an average fuel efficiency of 15 mpg.

40) Assume India & China (combined) have 3.5 billion people. Assume ⅛th of their population currently drive.

Suppose 1/5 th of the population were driving. What additional gasoline demand (in gal/day) would there be

assuming each driver averages 40 mi/day with an average fuel efficiency of 15 mpg?

41) Estimate the time it takes for a signal from Earth to reach a Jupiter probe. Use T = D/c and the speed of

light, c ≈ 3.0 × 108 m/sec. The distance from the Earth to Jupiter , D ≈ 6.3 × 10

11 m.

42) Estimate the flow in the Columbia River in gallons/day. Use a flow of 1.5 × 105 cfs and 7.48 gal/cu-ft.

43) Estimate the value of 1 billion pounds of gold. Use a value of $400/oz and 16 oz per pound.

44) Estimate the weight of $1 billion of gold. Use a value of $400/oz and 16 oz per pound.

45) Each ⅛ th inch of a phone book's white pages contains about 100 pages with each page having 3 columns.

On average, there are 10" of text per page with 11 phone numbers per inch. Estimate the number of phone

numbers in 2 ¼" of white pages.

46) There were approximately 6,400,000 buildings damaged by hurricane Kipp. A typical building repair will

use a crew of 5 working 40 hr/wk for 16 weeks. Estimate the man-hours needed to clean up after this

hurricane.


1.7 Absolute Value

Since we want to be able to represent forward vs. backward or up vs. down it seems natural to use

positive and negative numbers. However, as you can imagine, we are often simply interested in a

distance or size regardless of the orientation (sign). For instance, if we are building a 50' long wall, it

does not matter much whether we start at the right end and measure towards the left or begin at the left

end of the wall and measure towards the right.

Measuring the distance a number 'k' is offset from 0 on the

number line is called the absolute value of k and is denoted

by │k│. Hence, the absolute value of negative 4 is positive 4

or | -4 | = 4.

fig 1 The Absolute Value

When driving 65 mph in a 20 mph zone it really makes no difference whether we are traveling north or

south. In this case Smoky doesn't care about the sign of the number just its magnitude (size).

Another way to think of the absolute value of a number is the magnitude

(size) of the number. Algebraically, if x is non-negative (x ≥ 0) we make no

change, but if x is negative (x < 0) we change the sign of x to make the result

positive.

, 0| |

, 0

x when xx

x when x

Since absolute value brackets imply a grouping they imply parentheses and carry the same priority.

Thus, | expression | → | (expression) |. With | expression | it is important to reduce the entire expression

to a single value before applying the absolute value.

From a practical standpoint, to compute the absolute value of an expression, completely simplify the

expression to a single number and then strip away the negative sign if there is one and make the number

positive.

Note: A useful application of absolute value occurs when computing a

distance between two points on the number line where 'b' is not always

larger than 'a'. a b

length = a – b = b – a

fig 2 Distance Between a and b

Beware!

Absolute value notation can easily be mistaken for a "1" and care must be taken to avoid this

confusion. Absolute value does NOT change subtraction to addition. Any expressions must be

simplified using standard order of operations first before applying any absolute values.

Example 1 Simplify each expression

1) | -3 | = 3 2) | 8 | = 8

3) | 257 | = 257 4) | -0.06 | = 0.06

5) | -3½ | = 3½ 6) | 4.6 | = 4.6

7) | -14 | – 6 = +14 − 6 = 8 8) | 12 – 17 | = | -5 | = 5

9) 2·| 8 – 14 | = 2·| -6 | = 2·(+6) = 12 10) | 4(5 − 7) − 22 | = | -30 | = 30


Practice

1) | -7 | 2) | 5 | 3) | -13 | 4) -2| -7 |

5) | 24 – 35 | 6) -2 | 5 – 13 | 7) | 11 – 7 | 8) | 13 – 18 | 4 – 3

9) | 2 – 10 | 4 – 18/2 10) | 4.43 − 7.32 |

Answers-

1) 7 2) 5 3) 13 4) -14 5) 11

6) -16 7) 4 8) 17 9) 23 10) 1.7

1.7.1 The Absolute Value on the TI-83

There are two ways to compute absolute values using the TI-83.

(a) One choice is to apply the absolute values manually. To apply the absolute value manually,

simplify the expression within the absolute value brackets and then apply the absolute value by

changing the sign of a negative result. This can be done using the negative key. Positive results

can be left alone.

(b) The other choice is to use the built-in absolute value operator abs( ). Use abs ( expression ) in

the same way you use ( expression ). To find abs ( ), use MATH, NUM or select it from

CATALOG. Be sure to close the parentheses.

Example 2 Compute 24.6 − 15.8 | 32.8 − 62.4 |

(a) Manual Application

Compute the expression 32.8 − 62.4 → -29.6

Apply the absolute value to -29.6 by changing the sign of the answer

Complete the calculation

(b) Using Built-in abs ( ) You can use CATALOG to find abs( or MATH and NUM.

Key-in the entire calculation at once.

1.7 Exercises - Order of Operations with Absolute Value

Simplify by eliminating absolute value signs.

1) | -14 | 2) | 10 | 3) | -4 | 4) | 8|

5) | 7.6 | 6) | -9.7 | 7) | -12.4 | 8) | 29.1 |

9) | 5 ⅔ | 10) | -8 ¾ | 11) | -½ | 12) | ⅜ |

13) 2 · | 8 − 14 | 14) 3 · | 10 − 10 | 15) 12 · 8 ÷ | -4 | 16) | -1 − 9 | · 10

17) | 27 − 53 | 18) | 2 − 8 | 19) 10 − 5 | 10 − 25 | 20) 100 − 5 | -96 − 4 |

21) | -3 |2 22) 5 | 2

3 | 23) | -81 | 24) | 2 − 2 |

25) Use absolute values to write an expression for the distance between x1 and x2.

26) Use absolute values to write an expression for the distance between h and k.


1.8 Reading a Ruler or Tape Measure

Of all the measuring devices that you may use in your careers the two most common are the ruler and

the tape measure. Rulers and tape measures are commonly formatted with USCS units or metric units.

USCS devices are usually segmented either by feet, inches and fractional-inch increments, inches and

10th

-of-an-inch increments or feet with 10th

-of-a-foot increments. Metric devices usually contain meter,

centimeter and millimeter increments.

1.7.1 The Standard Ruler/Tape Measure

Rulers using feet and inches usually divide the inch up into repeated half divisions. Most commonly

there are distinctive marks at 1

2 , 1

4 , 1

8 and 1

16 inch.

Standard Ruler 1" 2"

116

18 1

412

58

1316 1 3

16 1 34

this ruler is not drawn actual size

Note: When we have a measurement of say 10/16th

inch but give it as 5/8th

inch we are giving the

impression that the measurement might only be accurate to the nearest 8th

inch. That is, we give the

impression we used a device where the smallest divisions were ⅛" rather than 1/16". Thus, we might

choose to use the non-reduced form of a fractions to indicate that the measurement is actually accurate

to the nearest 16th

inch.

Notice that we can reasonably estimate the next division beyond the 1/16" divisions shown. Here, we

can realistically estimate point A, B and C below. Note: We DO NOT say one half of a 16th

; we

convert to the equivalent in 32nds

.

B=1 1

32"A= 132" A=1 15

32"

1" 2" this ruler is not drawn actual size

Surveyors, in particular, use tape measures divided into 10ths

-of-a-foot. We shall refer to this as the

Surveyor’s Tape Measure. Notice that we can realistically expect to accurately estimate measurements

halfway between the tick marks. Here, we can realistically estimate point A, B and C below. Note: We

DO NOT say one half of a 10th

; we convert to the equivalent 0.05 increment.

B=1.55' C=3.15'

2' 1' 3' .5 .5 .5

A=0.05'



1.8.3 The Mechanical Ruler

A mechanical ruler is similar to a surveyor’s tape measure. However, the primary divisions are inches

with sub-increments of 1/10th

of an inch. Notice that we can realistically expect to accurately estimate

measurements halfway between the tick marks. Here, we can realistically estimate point A, B and C

below. Note: We DO NOT say one half of a 10th

; we convert to the equivalent 0.05 increment.

A=0.05" C=2.35"B=0.75"

.5 1" .5 2" .5 3"


1.8.4 The Metric Ruler/Tape Measure

The metric ruler typically has primary increments of centimeters and sub-increments of millimeters. It’s

not unusual to have a meter stick also marked with decimeters. Millimeters are small enough that half

millimeters might be difficult to distinguish.

2.25 mm

1

0.5 mm

2 3

1.0 cm

1.3 cm

13 mm

1 cm 3 mm


1.8.5 The Scaled Ruler (Architectural Ruler)

Scaled rulers are used to draw accurate scaled drawings. They are quite frequently used in architectural

or mechanical drawings. A common scaling for drawing a small building on 8½" x 11" paper is to use a

1' = 1/8" scale. We commonly refer to this as a 1/8" scale. With a 1/8" scale, a 7 ft wall would be

represented by a 7/8" long line drawn on the paper. Similarly, a 1 ⅝" (13/8") line on the paper would

represent a 13 ft wall. Often these rulers take one segment and divide it up among 10 or 12 sub-

increments.

1" 2" 3" 4"

10 20 301/8 Scale

Standard Ruler

78"

7 13

1 58"


Example 1 Give the equivalent measurement bases on ⅛" = 1'

(a) A 5

8 " drawn line represents 5'. (b) A 1

7

8 " ( 15

8 " ) drawn line represents 15'.

(c) A 11

16 " (

5 ½

8 " ) drawn line represents 5 ½ '. (d) A 2

11

16 " (

21 ½

8 " ) drawn line represents 21 ½'.

(e) 6' is drawn as a 6/8" (or ¾") line. (f) 17' is drawn as a 17/8" (or 2 ⅛") line.

(g) 3 ½' is drawn as a 3 ½

8 " (or 7/16") line. (h) 27.5' is drawn as a

27.5

8 " (or 3

7

16 ") line.


1.8 Exercises -

1) Fill in the remaining measurements

1" 2"

A=1 132"


B=1.55'

2' 1' 3' .5 .5 .5

0.2'

3) Fill in the remaining measurements 0.75"

.5 1" .5 2" .5 3"


1

2 mm

2 3

1.0 cm

1.3 cm

13mm

1 cm 3 mm

5) Use the ⅛" scaled ruler to answer the following questions.

1" 2" 3" 4"

10 20 30Arcitechural Ruler 1/8 Scale

Standard Ruler

(a) How long a wall does a 2 ⅜" drawn line represent?

(b) How long a wall does a 2 5/16" drawn line represent?

(c) How long a line should we draw to indicate a 18' wall?

(d) How long a line should we draw to indicate a 21.5' wall?


1.9 Rounding, Accuracy and Significant Digits

Decimal Place Names M

illi

on

s

Hu

nd

red

Tho

usa

nd

s

Ten

Tho

usa

nd

s

Th

ou

san

ds

Hu

nd

red

s

Ten

s

On

es

Dec

imal

Po

int

Ten

ths

Hu

nd

red

ths

Th

ou

san

dth

s

Ten

Tho

usa

nd

ths

Hu

nd

red

Tho

usa

nd

ths

Mil

lion

ths

1, 2 3 4, 5 6 7 . 8 9 0 1 2 3

1.9.1 The Standard Rounding Rule for Decimals

The standard rounding rule is to round up if the digit immediately to the right of the rounding digit is 5

or greater and to round down (truncate) if it is 4 or less. For example, to round 34.6847 to the nearest

hundredth (2nd

decimal place) we look at the digit immediately to the right in the thousandths place.

Since a 4 is immediately to the right of the hundredths place we round down (truncate) to get 34.68.

Note We DO NOT round sequentially. That is, we CANNOT first round up the 4 due to the 7 to its

right creating 34.685 and then round up the 8 due to the 5 which we got from rounding the 4.

CORRECT: 34.6847 ≈ 34.68 INCORRECT: 34.6847 ≈ 34.685 ≈ 34.69

Example 1 Round to the nearest…

hundredth 42.899 ≈ 42.90 hundred 358 ≈ 400

hundredth 4.5008 ≈ 4.50 thousand 404,399 ≈ 404,000

integer 3.14 ≈ 3 million 94,499,999 ≈ 94,000,000

half 42.36 ≈ 42.5 = 42 ½ quarter 63.83 ≈ 63.75 = 63 ¾

1.9.2 The Standard Rounding Rule for Fraction → Fraction

To round one fraction to another fraction we first strip off any whole part. Then create an equivalent

fraction for the rounded fraction by setting up a proportion. See § 1.4.2. The numerator is then rounded

to an integer by the same rule as for decimals above.

Example 2 Round 7 5

32 to the nearest ⅛th

.

Step 1 round 5

32 → x

8 Strip off the whole part

Step 2 x

8 =

5

32 Set up a proportion for an equivalent fraction

Step 3 x = 5 · 8

32 =

5

4 = 1.25 Solve for x by cross-multiplying

Step 4 1.25 ≈ 1 Round accordingly

Thus, 7 5

32 ≈ 7

1

8


Example 3 Round 6 59

64 to the nearest ¼

th.

Step 1 round 59

64 →

x

4 Strip off the whole part

Step 2 x

4 =

59

64 Set up a proportion for an equivalent fraction

Step 3 x = 4 · 59

64 =

59

16 = 3.69 Solve for x by cross-multiplying

Step 4 3.69 ≈ 4 Round accordingly

Thus, 6 59

64 ≈ 6

4

4 = 7

1.9.3 Fraction → Rounded Decimal

To round a fraction to a decimal, first divide the fraction to obtain a decimal form. Then round by the

same rule as for decimals above.

Example 4 Round 59

64 to the nearest thousandth.

Step 1 59

64 = 0.921875… Divide the fraction and underline the desired digits

Step 2 0.921875… An 8 just to the right of the thousandths place indicates we round up

Thus, 59

64 ≈ 0.922

1.9.4 Decimal → Rounded Fraction

It is very common to want an equivalent or rounded fraction for a given decimal value. For instance,

suppose we have a 60" tube and we must snip it into 7 equal pieces. Then each piece is 60"

7 ≈ 8.5714"

in length. But how can we realistically use a tape measure (with divisions in 16ths

inch) to mark the

cuts? We convert the decimal to a rounded fraction to the nearest 1/16th

.

To Round a Decimal to the Nearest Mixed Number Step 1 Strip off the whole part. Only the decimal part is converted to a fraction.

Step 2 Use a proportion to write the decimal as an equivalent fraction.

Step 3 Solve the proportion by cross-multiplying.

Step 4 Round the result to the nearest whole number.

Step 5 Put it all together.

Example 5 Round 8.5714" to the nearest 16th

0.5714 Strip off the whole inches.

0.5714

1 =

x

16 Use a proportion to write the decimal as an equivalent fraction.

x = 16 · 0.5714 = 9.1424 Solve the proportion by cross-multiplying.

9.1424 ≈ 9 Round the result.

8.5714" ≈ 8 9/16" Put it all together.


Example 6 Round 9.3569" to the nearest 64th

0.3569 Strip off the whole inches.

0.3569

1 =

x




9.3569" ≈ 9 23/64" Put it all together.

Example 7 A 30" bar is cut into 7 equal pieces. What is the length of each to the nearest 16th

inch?

30" / 7 = 4.2857…" Compute the length in decimal

0.2857… Strip off the whole inches.

0.2857

1 =

x




4 5/16" each Put it all together.

Practice Round to 8

ths Round to 16

ths Round to 64

ths

a) 5.389 ≈

d) 3.087 ≈

b) 12.0389 ≈

e) 8.9187 ≈

c) 2.9832 ≈

f) 0.00035 ≈

g) A 47" bar is cut into 3 equal pieces. What is the length of each to the nearest 16th inch?

Answers-

a) 5 3/8

d) 3 1/8

b) 12 1/16

e) 8 15/16

c) 2 63/64

f) 0/64

g) 15 11/16" each

1.9.5 Alternate Rounding Procedures

Truncating

Can you think of a situation where we would round down (truncate) even when we are more than

halfway to the next higher value? That is, can you think of a valid reason we might want to round 26.86

to 26.8 instead of 26.9?

Suppose we require a brass fitting to pass through a 26.86 mm hole. Suppose we can only purchase the

tubing in 0.1 mm increments. Our choices are either 26.8mm or 26.9mm. 26.8mm would fit slightly

loose in the hole and 26.9 would not fit at all. Thus, we should round down (truncate) even though it is

not the closest value.

Rounding Up

Can you think of a situation where we would round up even when we are less than halfway to the next

higher value? That is, can you think of a valid reason we might want to round 28.4 to 29 instead of 28?


Suppose we are drilling a hole for a bolt that is 0.443" and we want the bolt to fit without forcing it into

the hole. Assuming we have drill sizes in 64ths

inch increments, we would automatically round up to the

next higher 64th

to assure its fit. Why? 0.443" ≈ 28.4

64 " ≈

28

64 . But 28

64 "

would be too tight so we must

round to 29

64 ".

Odd/Even Rounding

Suppose when collecting field data the data collector is not the data entry person. The data collector

carefully records tree heights to the nearest half-foot. Unfortunately, the data entry person can only

enter whole numbers of feet. Thus, the data entry person must round every entry of the form XXX.5 and

they would normally be rounded up.

When we round we are implicitly expecting the differences to somewhat balance out. However, this

would not happen here. Rounding every XXX.5 upward would skew the entire data set upward. Thus,

we have a rule (The odd/even rounding rule) where we round only those numbers that end exactly with a

"5" to their nearest even neighbor. That is, 7.5 → 8 but 6.5 → 6.

Example 8 Round using the odd/even rule

1) Round to the nearest thousandth 13.2695 → 13.2695 → 13.270

2) Round to the nearest hundredth 4.825 → 4.825 → 4.82

3) Round to the nearest hundredth 4.8255 → 4.8255 → 4.83

4) Round to the nearest thousand 370,500 → 370,500 → 370,000

5) Round to the nearest thousand 360,505 → 360,505 → 361,000

Note: One could easily argue that if skewing the data is of great concern then perhaps rounding should

be reduced or eliminated altogether.

Practice 1) Round to thousandths: 56.0583 ≈ 67.0098 ≈

2) Round to hundreds: 862,084 ≈ 34,959 ≈

3) Round to nearest 16th:

47

64 ≈ 0.373 ≈

4) There’s enough money to buy 3.8 hot dogs. How should we round our order?

5) We need 5.2' of wire which is only sold by the foot. How should we round our order?

6) Compute the average of the following data points: 5.3, 4.7, 2.5, 7.5, 8.1, 8.5, 3.5, 4.5

7) Round each pt of problem (6) to the nearest 0.5 and compute the average of the result.

8) Round each pt of problem (6) to the nearest integer and compute the average of the result.

9) Round each pt of problem (6) by the odd/even rule and compute the average of the result.

Answers-

1) 56.058, 67.010

2) 862,100, 35000

3) 12/16, 6/16

4) 3 hot dogs

5) 6 ft

6) 5.575

7) 5.5, 4.5, 2.5, 7.5, 8.0, 8.5, 3.5, 4.5; 5.5625

8) 5, 5, 3, 8, 8, 9, 4, 5; 5.875

9) 5, 5, 2, 8, 8, 8, 4, 4; 5.5


1.9.6 Accuracy in Rounding and Significant Digits

Suppose Kate and her daughter measure a wall. One of them very carefully aligns her end of the wall

with the tape measure while the other aligns hers rather carelessly. What would you expect to be the

overall accuracy? Obviously not very good!

Thus, the final accuracy in any calculation should depend upon the least accurate value used anywhere

in the calculation.

What we mean by least accurate depends on how we are using the value. When we give a measurement

such as 52.4" we are implying the measurement is accurate to the nearest tenth of an inch (0.1"). The

possible error is ± 0.05". If the measurement was more accurate, say 52.40", then we should specifically

include the “0” in the hundredths place to indicate that we have accuracy to the nearest hundredth.

Beware!

When we write a value, the potential error due to standard rounding is

always considered to be half of the final digit. Thus, 7.3" implies a

range 7.25" − 7.349̄" (or 7.3 ± 0.05"). Similarly, 67.375 mm =>

67.375 ± 0.0005 mm and so on. Note: When we use the xx.x ± 0.05

format as in 7.3 ± 0.05" we ignore the fact that 7.35" would

traditionally be rounded up.

7.2 7.47.3

-0.05 +0.05

possible error

min=7.3-.05=7.25

max=7.3+.05=7.35

With fractions we follow the same reasoning as with the decimals above. 153

8 " is implied to be accurate

to the nearest 8th

inch. If the measurement was more accurate, say to the nearest 16th

as in 156

16 ", then

we should leave the fraction in 16ths

to indicate that we have accuracy to the nearest 16th

inch.

Beware!

As with decimals, when we write a fraction of the form a

b , the

potential error due to standard rounding is always considered to be

half of the fraction increment (i.e. 1

2b ). So, if we use 8

ths we imply

the error could be as much as 1/16th. In other words, 123

8 " implies a

range 12 5

16 " − 12 7

16 " (or 123

8 ± 1

16 "). Similarly, 183

4 lbs → 183

4 ± 1

8

lbs and so on. Note that we ignore the fact that 12 7

16 or 187

8 lbs

would traditionally be rounded up.

28

48

38

- 116 + 1

16

possible error

min = 38 - 116 = 5

16

max = 38 + 116 = 7

16

If we are combining a group of measurements with known errors then the expected measure and range

of error is easily given. We simply combine the expected values and combine the possible errors. That

is, total positive error is the sum of all positive errors and total negative error it the sum of all negative

errors.

Example 9 Estimate (1.25 ± 0.003") + (2.36 ± 0.004") + (2.4 ± 0.01")

(1.25" + 2.36" + 2.4") ± (0.003" + 0.004" + 0.01") = 6.01 ± 0.017"


Perhaps you have already thought of the following dilemma. Suppose we weigh an I-beam to the

nearest 100 lbs and get 75,000 lbs. If we write 75,000. lbs the included decimal point implies accuracy

to the nearest pound. But if we write 75,000 lbs we are implying accuracy only to the nearest thousand

pounds. There are two simple solutions to this dilemma. We can use an underscore to indicate which

digits should be considered accurate or we can write the value with its potential error explicitly

expressed.

Example 9 75,000 indicates that only the zeros in the ones and tens place are questionable. This measurement is

accurate to the nearest hundred.

75,000 ± 50 lbs implies the zeros in the ones and tens place are questionable. This measurement is

accurate to the nearest hundred.

When a number is written in scientific notation (p × 10±n

) all the digits of "p" are considered accurate.

Conversely, only accurate digits should be included in "p" when writing a decimal number in scientific

notation.

Example 10

(a) 75,000 → 7.50 × 104 (b) 6.30 × 10

6 → 6,300,000

(c) 0.00230 → 2.30 × 10-3

(d) 9.300 × 10-7

→ 0.0000009300

Example 11 Round and indicate the implied accuracy in the result. 1) Round to the nearest thousand: 132,695 → 133,000

2) Round to the nearest million: 4,825,000 → 5,000,000

3) Round to the nearest ten: 4,799.5 → 4,800

4) Round to the nearest thousand 379,876 → 380,000

5) Round to the nearest thousand 300,205 → 300,000

All measurements have inherent error. When we add or subtract measurements the overall error can be

exacerbated. This is also true when we multiply, divide or make more complicated calculations

involving measurements. First let’s look at what occurs when adding or subtracting measurements.

1.9.7 Measurement Rounding Conventions with Addition and Subtraction

When we add 245.69 + 137.2 we are really adding 245.69 +

137.2?. Thus, the hundredths digit in the result is

questionable and should be omitted by appropriate rounding.

245.69

+ 137.2?

382.89 ← questionable digit

≈ 382.9

When we add 17,000 + 1650 we are really adding

17,??? + 165?. Thus, The hundreds digit in the

result is questionable and should be adjusted by

appropriate rounding.

17,???

+ 1,65?

18,650 ← questionable digits

≈ 19,000 ← underline accurate digits

The Rounding Rule for Addition and Subtraction

Generally speaking, when addition or subtraction are used in a calculation involving

approximate values the final accuracy is determined by the least accurate measurement in the

string and the final result is rounded to the same accuracy as the least accurate measurement.


Example 12

1) 948.3567' + 137.5' − 12.36' = 1,073.4967' ≈ 1,073.5'

The least accurate measurement, 137.5', is assumed accurate only to the nearest tenth foot so its

hundredths digit is questionable (it could contain up to 0.05' of error).. Thus, the final answer's

hundredths digit is questionable and the result should be rounded to the nearest tenth.

2) 12,680' + 13,500' – 944' − 5,100' + 2,582' = 22,718' ≈ 22,700'

The least accurate measurements, 13,500' and 5,100', are assumed accurate only to the nearest

hundred feet so their hundreds digit is questionable (it could contain up to 50' of error). Thus, the

final answer's hundreds digit is questionable the result should be rounded to the nearest hundred.

Note: The above rule is simply a guideline. Although we round to the nearest tenth we certainly cannot

absolutely guarantee that digit. If we want a guaranteed accuracy in the combined result we may need to

expand the accuracy requirements of individual measurements beyond that place.

Beware!

Always keep full accuracy in all intermediate calculations. DO NOT round early. Only round at the very

end of the calculation.

Correct! 2.48 + 1.36 + 2.0 = 5.84 ≈ 5.8 Wrong! 2.48 + 1.36 + 2.0 ≈ 2.5 + 1.4 + 2.0 = 5.9

Similarly, in a string of additions and subtractions involving fractions, the final accuracy can certainly

be no more than the least accurate measurement in the string.

Example 13

1) 23 ⅝ " + 16 ½" = 40 ⅛" ≈ 40 0

2 "

The least accurate measurement, 15 ½", is only assumed accurate to the nearest half inch so it

could contain up to ¼ inch of error. Thus, the final result could also contain at least ¼" of error and

should be rounded to the nearest ½ ".

2) 10 6

8 " + 9

3

16 " − 15

1

4 " +

48

64 " =

348

64 " =

87

16 " =

21.75

4 " ≈

22

4 " = 5

2

4 "

The least accurate measurement, 15¼", is only assumed accurate to the nearest quarter inch so it

could contain up to ⅛ inch of error. Thus, the final result could also contain at least ⅛" of error and

should be rounded to the nearest ¼".

Practice Circle the least accurate measurement in the string. Then compute and round accordingly.

1) 19.405" + 12.30" − 16.254" + 8.001" = ≈

2) 7.005 mm + 4.230 mm − 19.222 mm + 1234 mm = ≈

3) 17 5

8 " + 29

13

16 " − 12

1

4 " + 16

13

64 " = ≈

Answers-

1) 12.30", 23.452" ≈ 23.45" 2) 1234, 1226.013mm ≈ 1226mm 3) 12¼", 51 25/64" ≈ 51 2/4"


1.9.8 Measurement Rounding with Multiplication, Division, Roots and Powers

Consider the area computation for a 3.6' × 24.2' rectangle.

The minimum size we could have is ≈ 3.55' × 24.15' with A ≈ 85.73 ft2

The likely size we would have is ≈ 3.6' × 24.2' with A = 87.12 ft2

The maximum size we could have is ≈ 3.65' × 24.25' with A ≈ 88.51 ft2

3.65

3.55

24

.25

24

.15

.05 .05

3.6

24

.2

Notice that the actual area could legitimately fall anywhere in the range 85.73 ft2 ≤ A ≤ 88.51 ft

2. Thus,

although each measurement is accurate to the nearest tenth the result is certainly not that accurate. In

fact, we are not even sure of the one’s place.

Consider a row of blocks where each block is 12.0". Using our addition/subtraction rounding rule for

two blocks yields 12.0" + 12.0" ≈ 24.0". If we had 100 blocks it’s likely some would be a little big and

some would be a little small so the error should tend to even out. But is it reasonable now to expect an

overall accuracy to the nearest 10th

inch? i.e. 12.0" + 12.0" +… + 12.0" ≈ 1200.0". Suppose, however,

that every block is short by 0.05". That is, each block measures 11.95". In that case 100 blocks would

measure 100(11.95") = 1,195" and generate an error of 100(0.05") = 5". From this example it should be

clear that multiplication can exaggerate a potential error.

Suppose we have a 52" tube and we cut it into 7 pieces. We should not expect each piece to be precisely

7.428571429" long. The fact that the initial piece was only known to the nearest inch should prohibit

the resulting piece from being accurate to the nearest billionth inch. On the other hand, rounding to the

nearest inch as in the addition/subtraction case should also seem unreasonable.

From the above examples it should be obvious that the rule for expected accuracy when multiplication

(or division) is involved must be different than the rule used for addition or subtraction.

When we multiply 120 × 30 we are really multiplying

12? × 3?. We can see from this example that the

accuracy of the result is no longer place dependent but

rather dependent on the number of accurate digits in

the measures.

12?

× 3?

? ? ? ← questionable digits

3 6 ? ← questionable digits

3 ? ? ? ← questionable digits

The Rounding Rule for Multiplication and Division

Generally speaking, when multiplication or division are used in a calculation involving

approximate values the final accuracy is determined by the approximate value with the fewest

number of significant digits in the computation and the final result is rounded to the same

number of significant digits.

Before we can put this rule to use we must understand exactly what we mean by significant digits.

1.9.9 Significant Digits

Significant digits are any digits that are not used as placeholders. Since only zeros are used as

placeholders every non-zero digit is automatically a significant digit. Only zeros are in question. 6,000

could easily be an approximation for the number 5,128. The zero's of 6,000 are necessary placeholders.


13 ten-thousandths = 0.0013. These zero's are necessary placeholders. Putting a number in scientific

allows us to distinguish the significant digits from the zeros acting as placeholders.

± X . X X X X · 10 ± n

sign of number

non-zero leading digit

significant digits

integer exponent

direction for decimal point shift

Some Observations:

2,007 2,007 = 2.007 × 103. None of these zeros are placeholders. All 4 digits are significant.

3,500

3,500 can be assumed to be accurate only to the hundreds place. When we refer to a "thirty

five hundred pound truck" we could easily be referring to a truck weighing 3,522 lb. Thus,

these 0’s are not guaranteed to be accurate they are only place-holders. As written, 3,500 has

2 significant digits.

0.00015 0.00015 = 1.5 × 10-4

. These zeros are all placeholders. There are only 2 significant digits.

2.30 × 105

2.30 is implied accurate to the hundredth's place. Such zeros represent accuracy not place-

holders and are significant digits. We have 3 significant digits and would write 230,000.

Which Digits are Significant?

Any non-zero digit is automatically significant. 1,234.56 has 6 significant digits.

Any zero between non-zero digits is significant. 100.2 has 4 significant digits.

If the value is a whole number (decimal point not included) then all trailing zeros are placeholders

and not significant. Any such zeros should be underscored if they are significant. 3,000 has 1

significant digit, 24,000 has 2 significant digits, 43,750 has 4 significant digits. 93,000 has 5

significant digits.

If the value is a whole number and its decimal point is included then all zeros are significant.

However, the underscore is a better way to indicate significant zeros. 3,000. has 4 significant digits.

If the value is less than 1, any leading zeros are non-significant (they are placeholders) but any

trailing zeros are significant. 0.01 has one significant digit, 0.0012 has 2 significant digits, 0.0050

has 2 significant digits, 0.00025 has 2 significant digits. 0.200 has 3 significant digits.

A cardinal number (exact count) has unlimited significant digits. When we say 4 cars we know there

were exactly 4 cars, not potentially 4.1 cars.

In short, every zero digit is significant except the trailing zeros of whole numbers and leading zeros for

numbers less than 1. When zeros appear after the decimal point for numbers greater than one then they

are signigficant.

123,000,000trailing zeros

non-significant

123,000,000trailing zeros

non-significant

0.000123leading zeros

non-significant

1000.000all zeros aresignificant

Example 14

(a) 3,289" has 4 significant digits (b) 1,001" has 4 significant digits

(c) 1,200" has 2 significant digits (d) 0.0015" has 2 significant digits

(e) 150.0" has 4 significant digits (f) 4 dogs has unlimited significant digits


(g) 200" has 3 significant digits. The underscore indicates the significant zeros

(h) 7 dogs This is an exact count and has an unlimited number of significant figures.

(i) 1,000 dogs

Here we must consider this as a potentially rounded number. Only the "1" is

assumed to be significant unless told otherwise. Suppose we want to indicate

that 1,000 dogs is an exact count. What could we do? How about 1,000 dogs

Practice Give the number of significant digits for each measurement

1) 40.25 mm 2) 1600 # 3) 12,000 T 4) 0.0034" 5) 1.002 cm

6) 8 cats 7) 500 bolts 8) 100 bolts 9) 3.50 × 105 m 10) 6.8 × 10

-5 hz

Answers-

1) 4 2) 2 3) 3 4) 2 5) 4

6) unlimited 7) 1 8) unlimited 9) 3 10) 2

1.9.10 Applying the Multiplication/Division Rounding Rule

Remember, the number of significant digits in the final result is determined by the worst-case scenario.

The measurement with the fewest significant digits controls the final outcome by restricting the result to

the same number of significant digits as in the worst intermediate measurement.

Example 15 Find the area for a 24.2' x 3.6' rectangle and round accordingly.

Since there are three significant digits in 24.2' but only two significant digits in 3.6' we can expect no

more than two significant digits in the final result.

A = (3.6')(24.2') = 87.12 ft2 ≈ 87 ft

2

Notice that using two significant digits is still slightly optimistic since the area could still fall anywhere

inside the range 85.73 ft2 ≤ A ≤ 88.48 ft

2. Why?

Example 16 Use 7.48052 gal = 1 cu-ft to approximate the volume of a 55 gal barrel in cubic feet.

Since 55 gal has only 2 significant digits the result is rounded to 2 significant digits.

V ≈ 55 gal × 1 cu-ft

7.48052 gal = 7.352…cu-ft ≈ 7.4 cu-ft.

Example 17 Approximate the total weight of 1,500 cans. Each can weighs 32.4 oz

Although the 1,500 cans is a count as written it's an approximate measurement. So the fewest number of

significant digits are the two digits of 1,500 cans. Thus we should round the result to 2 significant digits.

W ≈ (1,500 cans)(32.4) = 48,600 oz ≈ 49,000 oz.

Just as we can estimate the final accuracy of a computation we can estimate the accuracy needed in the

measurements used to achieve a predetermined accuracy. In such a case, perform the computation with

rough estimates. This tells us the magnitude of our result. Count the number of significant digits we

want in the result. Then each measurement must be made to that level of accuracy.


Example 18 What accuracy is required when measuring the length and width of a 157' × 293' rectangle to compute its

total area to the nearest square foot?

First, estimate the area: A ≈ 157' × 293' = 46,001 ft2. 'To the nearest square foot' requires the one's

digit be accurate. Why? Second, count the significant digits needed in the result. In this case 5.

To have accuracy in the one's place will take 5 significant digits in the result. Thus, each measurement in

the calculation must have 5 significant digits. Hence, each measurement must be accurate at least to the

nearest hundredth foot to obtain 5 significant digits. i.e. 157.00' × 293.00'

Example 19 Suppose the area of Example 18 must be accurate to the nearest sq-in. How accurately must the length

and width be measured now?

Estimate the area in sq-in: 157' = 1,884", 293' = 3,516" A ≈ 6,624,144 in2.

'To the nearest square inch requires 7 significant digits needed in the result. Why?

Thus, each measurement in the calculation must have 7 significant digits. i.e. 1884.000" × 3516.000"

Example 20 What accuracy is required when measuring the radius of a 30" diameter circle so that the area will be

accurate to the nearest 0.1 sq-in?

First, estimate the area. A ≈ π (15")2 = 706.858… in

2. The nearest 0.1 sq-in implies the tenth's digit be

accurate. Why? Second, count the significant digits needed in the result (in this case 4).

To have accuracy in the tenth's place will take 4 significant digits in the result. Thus, each measurement

in the calculation must have 4 significant digits. Hence, the diameter (or radius) must be accurate at least

to the nearest hundredth foot to obtain 4 significant digits. i.e. 15.00" (or 30.00" dia)

Note: π's approximation must also contain at least 4 significant digits.

Practice Compute and Round Accordingly 1) Use 7.48052 gal ≈ 1 cu-ft to approximate the volume of a 500.0 gal tank in cu-ft.

2) Use 7.5 gal ≈ 1 cu-ft to approximate the volume of a 500.0 gal tank in cu-ft.

3) Use 1.609344 km ≈ 1 mile to approximate the distance 257 miles in kilometers.

4) Use 0.6 mi ≈ 1 km to approximate the distance 257 miles in kilometers.

5) What accuracy is required in the mi → km conversion to compute 37,968 km to the nearest mile?

6) What accuracy is required for the radius (and π) to approximate a circle's area accurate to the nearest sq-ft

when the diameter is approximately 100 ft?

7) What accuracy is necessary in the individual dimensions to obtain the area of a rectangle to the nearest sq-ft

if its dimensions are approximately 100' x 50'?

8) A 12% mix is made from 12cc of Lycane & 88cc of saline. What accuracy is necessary in each component

to expect the concentration to be accurate to the nearest 0.1%?

Answers-

1) 66.84 cu-ft

2) 67 cu-ft

3) 414 km

4) 400 km

5) 5 sig digits, 1.6093 km/mi

6) 4 significant digits

7) 4 significant digits,

base ± 0.1, ht ± 0.01

8) 0.1cc for each


1.9 Exercises - Rounding 1) Round to hundredths: 23.092165 2) Round to tenths: 23.092165

3) Round to tens: 23.092165 4) Round to thousandths: 43,982.0349821

5) Round to thousands: 43,982.0349821 6) Round to hundredths: 521.398777

7) Round to the nearest unit: 2300.215 8) Round to the nearest hundredth: 0.00213

9) Round 39

64 to the nearest 16

th. 10) Round

15

64 to the nearest 8

th.

11) Round 0.697 to the nearest 64th

. 12) Round 12.783 to the nearest 16th

.

13) James needs 2.2 quarts of paint. How should he round his purchase? Why?

14) Betty has collected enough money for 4.8 pizzas. How many should she order? Why?

15) Sam must drill a hole for an 8mm metric bolt (≈ 0.315") using a bit from his drill index consisting of bits in 64ths

.

What size bit should he use?

16) McKenzie must fit a keyway into a 44 mm slot (≈ 1.7323"). Her keyways are marked in 64ths

. What size should she

use?

17) Find the area to the nearest 100th

sq-in

13738"

area = ?51 716"

18) Find the diameter of each circle to

the nearest 64th

inch 0.2626"

0.4381"

0.6686"

Compute and round appropriately:

19) 15.67 + 13.2 20) 1,536 + 340 21) 824 + 1100 −17713

22) 9.4560 + 691.54 23) 602.36 − 2.364 24) 3289 + 2285 − 1600

25) 15.67 × 13.2 26) 1,536 × 340 27) 6.22·749

2.72

28) 9.4360 + 684.54 29) 9.002 · 2.00 30) 2,000

72

31) 93,000 × 0.02562 32) .000125 × 8,569 33) 0.0175 × .20 × 1.9755

34) 0.010 × 2793 35) 420,000 × 140.0 36) 1,000,000.00 π

37) π ÷ 0.1000 38) 3867 ÷ 0.967 39) 2,359 ÷ 0.00786

40) Circle the least accurate measurement in the expression then compute and round accordingly.

2.69" + 175" – 24.25" + 0.005"

41) Give the number of significant digits:

23.69 ft 54.20 m 230 cm 3,400. mi 0.0005 sec 2002 L 0.0200 lbs

42) What accuracy is required in each measurement to compute the

total area to the nearest sq-ft?

456'

62'

43) When a typical tape measure is used what accuracy is implied by:

6 15

16 " ± 9' 3¼ " ± 0.003 mm ± 48.2' ± 2700 m ±

44) Compute and round based on significant digits:

(a) P = π

0.1751 ≈

use π ≈ 3.14

(b) P = π

0.1751 ≈

use π ≈ 3.14159

(c) A = π 21.002

use π ≈ 3.14

(d) A = π 21.002

use π ≈ 3.14159


45) Compute and round based on significant digits:

(a) E = (440 lbs)(32.89 ft)

9.98 sec2 (b) A = (3.65")(0.28") + (11")(5")

46) Approximate the length of the plate and give the maximum

possible error. Give your answer as L ± e".

1.3 ± .02 " 2.6 ± 0.2" 1" ± 1/16"

r = 3/8 ± 1/64"

47) What accuracy is required in the radius measurement so that the total area of a circle is accurate to the

nearest square meter? The radius is approximately 60 m?

48) What accuracy is required (in inches) in the radius measurement so that the total area of a circle is accurate

to the nearest sq-ft. The radius is approximately 40'?

49) What accuracy (nearest fraction of an inch) is required in

each measurement to compute the total area to (a) the

nearest sq-ft, (b) the nearest sq-in? 37' 7"

22' 11"

Compute and round based on significant digits:

50) Approximate the total area of 5 steel plates. Each plate is 3.6" x 24.2"

51) Approximate the average speed when covering 51 feet in 40.28 seconds

52) Approximate the volume of the cube which is 14 cm per side.

53) Approximate the equivalent to 1,768.2 feet in kilometers. Use 1 km = 3280 ft

54) Approximate the area in a 15" diameter circle.

55) What accuracy is required for a 117" × 43" rectangle to compute its area to the nearest 0.1 sq-in?

56) What accuracy is required for a 8' 7" × 4' 3" rectangle to compute its area to the nearest sq-in?

57) What accuracy is required for a 8,240' × 5,350' rectangle to compute its area to the nearest sq-ft?

58) What accuracy is required for a 118' dia circle to compute its area to the nearest sq-ft?

59) What accuracy is required for a 3" dia bearing to compute its circumference to the nearest 0.001 inch?

60) A 10% epi solution is made from 90 mg saline and 10 mg pure epi. If the final solution must be 10.0%

what accuracy must be used in measuring the two components?

61) A 14% alcohol solution is made from 86 mg pure water and 14 mg pure alcohol. If the final solution

must be 14.0% what accuracy must be used in measuring the two components?

62) George measures a room as roughly 12' × 16'. Would 200 sq-ft of tile always be enough? Why?

63) Lindy weighs a bag of soil on an assembly line at 42 lb. If 10,000 bags are produced what is the greatest

shortfall that might occur?

64) If 3.14 is used to approximate π, what error is expected in a circle's area when the radius is 1000.000 ft?


1.10 Formulas and Substitution

We often use a formula to make a calculation. For instance, if we want to compute the perimeter of a

rectangle we use the area formula P = 2L + 2W. As long as we know L (the length) and W (the width)

we can evaluate the formula. We do this by substituting values in place of the variables. We should

also substitute units along with the values. It is important that substitutions be done exactly as indicated.

That is, substitutions are verbatim. When we substitute, we not only input the value, but we also must

input the units.

Example 1 Using P = 2L + 2W find the perimeter of a rectangle 7.0 ft × 4.0 ft rectangle

L is replaced by 7.0 ft and W is replaced by 4.0 ft

P = 2(7.0 ft) + 2(4.0 ft) = 14.0 ft + 8.0 ft = 22.0 ft → 22 ft

Example 2 Using C = πD give the circumference of a 30.00" diameter circle.

D is replaced by 30.00 ft and π is replaced by the most accurate π we have

C = π 30.00" = 94.2477796…" ≈ 94.25"

There are a variety of mathematical notations that can be used to indicate substitution. One abbreviation

is to use a vertical line with the indicated substitution written below the line.

Example 3

(a) P = 2W + 2H | → 2·3 + 2·5 = 16

W = 3, H = 5 notation indicates substitution for W and H

(b) C = π D | → π 10

D = 10 notation indicates substitution for D

(c) V = π R2 H | → π 7

2 H = 49 π H

R = 7 notation indicates substitution for R

Just as we can substitute values and units in an expression or formula, we can also modify a formula by

replacing a variable(s) with an equivalent expression.

Example 4

C = π D gives a circle’s circumference in terms of its diameter. D = 2R gives a circle's diameter in terms

of its radius. Use substitution to get a formula for the circumference in terms of the radius.

D is replaced by 2R. C = π D = π (2R) = 2π R

So, C = 2πR

In shorthand substitution notation

C = π D | → 2·3 + 2·5 = 16

D = 2R


Example 5

A = π R2 gives a circle’s area in terms of its radius. R = ½ D gives a circle's radius in terms of its

diameter. Use substitution to get a formula for the area in terms of the diameter.

R is replaced by ½ D. A = π R2 = π (½ D)

2 = π (½ D)(½ D) = π ¼ D

2. So, A = ¼ π D

2

Beware! Watch out for negative numbers mixed with subtractions during substitution.

Example

R = p – q; p = 16.2, q = -8.4 Correct R = 16.2 – (-8.4) Incorrect R = 16.2 − 8.4

Watch out for negative numbers mixed with exponents during substitution.

Example

H = a2 − b

2; a = 4, b = -2 Correct H = a

2 − (-2)

2 Incorrect H = a

2 − -2

2

The parentheses are absolutely necessary. Why?

To be safe, use parentheses when substituting anything other than a single positive number!

Example 6 Substitute as indicated

(a) Q = 5W + H2 | → 5(x + y) + (-k)

2

W = x + y, H = -k

(b) P = R2 − H | → (a − b)

2 − (-1)

R = a − b, H = -1

1.10.1 Application of a two-stage formula

Heron’s Formula allows us to find the area of a non-right triangle when

we know the length of the three sides.

First define s = a + b + c

2

Then compute the area, A = s(s − a)(s − b)(s − c)

c ba

Example 7 Find the area of this triangle

a = 12.2, b = 18.1, c = 21.9. s = 12.2 + 18.1 + 21.9

2 =

52.2

2 = 26.1

A = 26.1(26.1 − 12.2)(26.1 − 18.1)(26.1 − 21.9) = 12189.744 ≈ 110

12.218.1

21.9

Notice that if we were to substitute the expression a + b + c

2 in place of 's' in the root we get a horrendous

formula. Thus, keeping the formula in two parts makes it much easier to use. Note also, in this formula, it

makes no difference which side is chosen as a, b or c. Why?

1.10.2 Substitution with the TI-83

The TI-83 calculator is ideally suited to perform value substitution. First, store each substitute value in

its alpha character. Then type the formula using the alpha characters. This method allows us to avoid


some of the common arithmetic mistakes associated with substitution. Additionally, this method is

especially convenient when we use the same value in multiple substitutions.

Example 8 Substitute as indicated

(a)

A = π R2 | → π 5

2 ≈ 78.5

R = 5

5 STO ALPHA R

2nd

π ALPHA R x2

(b)

Q = 5W + H2 | → 5(6) + (-5)

2 = 55

W = 6, H = -5

6 STO ALPHA W

-5 STO ALPHA H

ALPHA W + ALPHA H x2

If a complex formula (e.g. Heron's formula ) will be used repeatedly, a programmable calculator such as

the TI-83+ is very convenient. Since we shall expand upon the programmable features of the TI-83+

throughout the text this is a good starting point.

Notes: = ENTER, use ALPHA to activate alpha characters, use 2nd ANGLE for ', = STO

A TI-83+ Program for Heron's Formula

COMMAND COMMENTS

Press PRGM Brings up the Program Menu: EXEC EDIT NEW

Select NEW Use EDIT to edit an existing program

Name=HERON Names the program HERON. Other names will also suffice.

:ClrHome PRGM → I/O → 8. Clears the home screen.

:Disp "HERON'S FORMULA" PRGM → I/O → 3. Will display on the screen. (optional)

:Prompt A,B,C PRGM → I/O → 2. Will prompt the user for A, B and C

:(A+B+C)/2 S Calculates 's' and stores it in S.

:√(S(S−A)(S−B)(S−C)) Q Calculates the root and stores it in Q.

:Disp Q PRGM → I/O → 3. Displays the result.

:End PRGM → CTL → 7. Sets an End-of-Program mark. (optional)

Now let's run the program. Use PGRM → EXEC → Select Program . Use the cursor or type the

program ID number to select the program. You must hit ENTER to actually execute the program.

Example 9 Use the TI program to solve Example 7

PGRM → EXEC → HERON


Practice Compute and round according to the rules of significant digits.

1) A = B · H

B = 12.5, H = 6.2

2) A = (½)[B · H]

B = 3.5 , H = 2.75

3) V = 1

2 (R

2 – r

2) H

R = 32.5, r = 3.5, H = 6.9

4) H = a2 − c

2

a = 25.0, b = 24.0

5) A = b1 + b2

2 · h

b1 = 7.2, b2 = 6.8, h = 12.5

6) s = (x̄ – xi)

2

n – 1

x̄ = 4.5, xi = 3.8, n = 65

Answers-

1) 77.5 ≈ 78

5) 87.5 ≈ 88

2) 4.8125 ≈ 4.8

4) 7 ≈ 7.00

3) 11,315.39 ≈ 11,000

6) 0.00765625 ≈ 0.0077

1.10 Exercises – Substitution I

Compute and round accordingly.

1) A = L · W (a) L = 2.8, W = 9.5 (b) L = 1½, W = 3¼; L, W exact

2) A = b1 + b2

2 h (a) b1 = 9, b2 = 3, h = 4 (b) b1 = 4.5, b2 = 3.8, h = 4.6

3) A = π R2 (a) R = 8.0 (b) R = 10 ½ exact

4) f = 1

2π

g

L (a) g = 32, L = 10 (b) g = 9.81, L = 6.21

5) v = k

m (x0

2 − x

2) (a)

k = 2.1, m = 0.10

x0 = 9.6, x = 4.0 (b) k = 6.3, m = 5.1, x0 = 78, x = 63

6) V = 4 π r

3

3 (a) r = 8.0 (b) r = 6.375

7) R = R1 R2

R1 + R2 (a) R1 = 24, R2 = 12 (b) R1 = 16.8, R2 = 12.4

8) f =

1

1

p +

1

q

(a) p = 10, q = 15 (b) p = 3½ , q = 8¼; p, q exact

9) C = 5

9 [F − 32] (a) F = 98.6 (b) F = -20

10) F = 9

5 C + 32 (a) C = 100 (b) C = -40

11) KE = m v

2

2 (a) m = 24, v = 0.25 (b) m = 0.010, v = 80

12) PE = ρgh (a) ρ = 1.50, g = 32.2, h = 104 (b) ρ = 0.0900, g = 9.807, h = 2000

13) E = k(x − x0) (a) k = 0.23, x = 24.2, x0 = 2.0 (b) k = 375.01, x = 49.09, x0 = 36.29


14) s = (x1 − x0)

2 + (x2 − x0)

2

n − 1 (a)

x0 = 16, x1 = 24, x2 = 18,

n = 32; n exact (b)

x0 = -23, x1 = 42, x2 = -43,

n = 27; n exact

15) SA = πrh + 2πr2 (a) r = 4.8, h = 8.2 (b) r = 29.78, h = 132.81

16) T = πR2 − πr

2 (a) R = 100, r = 80 (b) R = 100, r = 80

17) i = pr t (a) p = 8,000, r = 6%, t = 3;

r, t exact (b)

p = 9,862, r = 4.675%, t = 12;

r, t exact

18) P = 2L + 2W (a) L = 8½, W = 5¼;

L W exact (b)

L = 136⅓ , W = 192⅔;

L W exact

19) E = (½)I2 (a) I = 12.8, = 3.42 (b) I = 53.42, = -4.38

20) m = y2 − y1

x2 − x1 (a)

x1 = 24, x2 = 56,

y1 = 12, y2 = 36;

x1, x2, y1, y2 exact

(b)

x1 = -7, x2 = 6,

y1 = 2, y2 = -6;

x1, x2, y1, y2 exact

21) = + + (a) = 37, = 53, = 90;

, , exact (b)

= 13.35, = 21.6, = 164;

22) = L·F (a) L = 1.5, F = 60 (b) L = 2.0, F = 30

23) m = y

x (a)

y = 38, x = 4;

y, x exact (b) y = 4.2, x = 6.8;

24) D = b2 – 4ac (a) a = 2.9, b = 13.5, c = 2.4 (b)

a = 3, b = -2, c = -16;

a, b, c exact

25) x̄ = x1 + x2 + x3

3 (a) x1 = 98, x2 = 82, x3 = 88 (b) x1 = -3.2, x2 = 9.8, x3 = 7.5

26) a =

360 2πR (a) = 60, R = 80 (b) = 123.7, R = 185.27

27) (½) (a +b+c) = s

A = s(s − a)(s − b)(s − c) (a) a = 4.0, b = 6.0, c = 8.0 (b) a = 3, b = 4, c = 5; a, b, c exact

28) B = D

1 + P2 (a) D = 64, P = 0.12 (b) D=24 , P = -8.0%

29) a = c2 − b

2 (a) b = 12.5, c = 16.5 (b) b = 292, c = 419; a, b exact

30) V = πH

3 (r

2 + Rr + R

2) (a) H = 2.8, r = 1.6, R = 3.8 (b) H = 15.8, r = 7.9, R = 30.1

31) D = R T (a) R = 60, T = 2.00 (b) R = 60, T = 5.03

32) C = 2πR (a) R = 10; R exact (b) R = 4⅜; R exact

33) y = mx + b (a) m = 3/5, x = 20, b = -3;

m ,x, b exact (b) m = -1.6, x = -44.6, b = 87.5


34) C = Ax + By (a) A = 2, B = -4, x = -6, y = 3;

A, B, x, y exact (b)

A = -2⅓ , B = 5⅔ , x = 8¾ , y = -7⅜;

A, B, x, y exact

35) y = y0 + m(x − x0) (a) x0 = -40, x = 10, y0 = -12,

m = 1½; m exact (b)

x0 = -32, x = 17, y0 = -16,

m = -2⅜; m exact

36) B = N

1 + R (a) N = 12,000, R = 3%;R exact (b) N = 26,000, R = -5%;R exact

37) N = B + R B (a) R = 5.0%, B = 2,350 (b) R = -8.0%, B = 4,400

38) B = D P

1 + P2 (a) D = 5,280, P = -8.05% (b) D=5,280, P = 3.60%

39) y = x (a) x = 2; x exact (b) x = 2.0000

40) y = k π (a) k = 17; k exact (b) k = 17.0000


1.11 Some Geometry Basics

We will discuss Euclidean geometry in detail in later chapters. However, you should already be familiar

with the basic 2-D shapes of rectangles, triangles and circles as well as common 3-D shapes such as

rectangular solids, cylinders and cones. We review a few of these basic shapes here since they are used

in a variety of examples throughout the text prior to the in-depth geometry section. A more in-depth

discussion is found in Chapter 4.

There are two characteristics of plane geometrical figures that we almost always want to know, namely

an object's perimeter and its area. Perimeter = "peri" + "meter" = "around" + "measure". Thus, the

perimeter of a 2-D object is the distance around the object. It only makes sense to use the word

perimeter when we are restricted to a plane. For a solid object (3-D object) the perimeter can be

ambiguous.

The area of an object is the region it covers. Area has square units.

1.11.1 Rectangles

To find the perimeter of any straight-sided plane figure (rectangles included) we

add up the lengths of all the sides. We typically use P for perimeter. With a

rectangle, the opposite sides are of equal length. This allows us to condense the

perimeter formula to P = 2L + 2W.

The formula for area of a rectangle is Area = Length × Width. A = L·W

L

W W

L

P = 2L + 2W

A = L·W

Example 1 Find the perimeter and area of the following rectangles (a)

(b)

(c)

P = 2(10.6) + 2(12.4) = 46.0

A = (10.6)(12.4) = 131.44 ≈ 131

P = 2(5.4) + 2(8.6) = 28.0

A = (5.4)(8.6) = 46.44 ≈ 46

P = 2(13 ft) + 2(24 ft) = 74 ft

A = (13 ft)(24 ft) =

312 ft2 ≈ 310 ft

2

1.11.2 Triangles

To find the perimeter of a triangle add up the lengths of all the sides. We

typically use P for perimeter. In this example the perimeter, P = a + b + c.

c ba

All triangles can be shown to be exactly half of a rectangle. The base & height

of the triangle correspond exactly with the base and height of the rectangle.

Thus, the area of a triangle is half the area of a rectangle.

That is, Area = 1

2 × Base × Height or A =

B·H

2


Beware! Be careful when determining the base & height. The height is the perpendicular distance to the base. It is

not necessarily the distance along one of the sides. Additionally, the base need not be horizontal. The only

requirement is that the base and height be arranged at right angles.

Example 2 Find the perimeter and area of the following triangles (a)

(b)

(c)

P = 5.0 + 12.0 + 13.0 = 30.0

A = (½)(4.6)(13.0) =

29.9 ≈ 30

P = 3.5 + 7.0 + 4.0 = 14.5

A = (½)(3.5)(2.7) =

4.725 ≈ 4.7

P = 10.0 m + 6.0 m + 8.0 m =

24.0 m

A = (½)(8.0 m)(6.0 m) =

24.0 m2 ≈ 24 m

2

1.11.3 Circles

We commonly use the term circumference instead of perimeter for a circle.

We typically use C for circumference.

The diameter is the distance across the circle through its center and the radius

extends from the center to the perimeter.

For every circle, the diameter will fit approximately three times around the

circumference. That is, C ≈ 3 × diameter. Check it. Unfortunately it's not

exactly "3". In fact, that number cannot be exactly represented by any

decimal. So we use the special symbol, π (pi). π = 3.14159….

The circumference formula is C = π D. Since the diameter is twice as long as

the radius (i.e. D = 2R) another common formula for the circumference is

given by C = 2 π R.

The area of a circle is given by A = π R2.

diameter

rad

ius

circumference

D = 2R, R = ½D

C = D, C= 2R

A = R2

Example 3 Find the circumference and area of the following circles (a)

(b)

(c)

R = ½ 20.0 = 10.0

C = π (20.0) ≈ 62.8

A = (20.0)2 = 400 π ≈ 1260

R = (½)(63 ⅓ ft) = 31 ⅔ ft

C = π (63 ⅓ ft) ≈ 199 ft

A = (63 ⅓ ft)2 ≈ 12,600 ft

2

D = (2)(142.0 ft) = 284.0 ft

C = π (284 ft) ≈ 892.2 ft

A = (142.0 ft)2 ≈ 63,350 ft

2


Beware! It is common for students to switch the formula for a circle's area with its circumference formula. One

way to keep things straight is to remind yourself that the circumference has units of length (e.g. ft) while

the area has square units (e.g. sq-ft). Thus, the area formula must be the one with a squared factor and

the circumference cannot have a squared factor.

Beware!

When we compute a perimeter or a circumference we should be careful to use compatible units

of length. Similarly, when we compute an area we should use commonly accepted units of area.

Be careful if the units are mixed such as ft-in.

When we compute the area of a 1.5' x 6" rectangle we should NOT use A = 1.5' × 6". Why? Instead we

should use A = 1.5' ×0.5' = 0.75 sq-ft or A = 18" × 6" = 108 sq-in. When computing area DO NOT use

mixed units. Convert to either feet or inches depending on whether you prefer sq-ft or sq-in in the

answer.

Example 4 Find the perimeter and area of the following figures (a)

4' 9"

3' 10"

(b)

1' 8"

H = 3' 10" = 3(12") + 10" = 46" (as inches)

H= 3' 10" = 3 ft + 10

12 ft ≈ 3.833' (as feet)

B = 4' 9" = 4(12") + 9" = 57" (as inches)

B = 4' 9" = 4 ft + 9

12 ft = 4.75' (as feet)

P = 2(4' 9") + 2(3' 10") =

2(57") + 2(46") =

114" + 92" = 206" = 17' 2"

A = B H = 57" · 46" = 2,622 in2 ≈ 2,600 in

2

A = B H = 4.75' · 3.833' ≈ 18.2 ft2

D = 1' 8" = 1(12") + 8" = 20" (as inches)

D = 1' 8" = 1 ft + 8

12 ft ≈ 1.667' (as feet)

C = π 20" ≈ 63"

C ≈ π (1.667') ≈ 5.2 ft

R = (½)20" = 10" = 10

12 ft

A = (10")2 ≈ 314 in

2

A = [ 10

12 ft]

2 ≈ 2.18 ft

2

Practice 1.11.3 Find the perimeter and area of the following figures (a)

(b)

(c)

1' 8"

11"

Answers

(a) P = 286.0, A = 2906.64 ≈ 2910 (b) P ≈ 105 m, A ≈ 887 m2 (c) P = 62" = 5' 2", A = 220 in

2 ≈ 1.53 ft

2


1.11.4 Solids

Solids with plane faces are called polyhedrons. In the 1900's,

Buckminster Fuller (1895-1983) popularized the use of Geodesic

Domes (polyhedron surfaces) in a wide variety of construction

projects. Polyhedrons with a common cross-section between two

parallel faces are called prisms. Rectangular prisms (boxes) are very

common. The volume of a rectangular prism is given by:

Volume = Length × Width × Height

All prisms have a common formula for their volume. V = Abase × Ht. In fact, all solids with a common

cross-section between two parallel surfaces have the same volume formula. Note: The height is always

measure perpendicular to the base.

Some Solids with a Common Cross-Section

Rectangular Prism

Triangular Prism

Circular Cylinder

V = x y z V = Abase × h = (½) x y h V = Abase × h = π r2 h

Example 5 Find the volume of the following figures (a)

x = 102"

y = 68"

z = 51"

(b)

r = 24.8 cm

h = 45.4 cm

V = x y z = 102" × 46" × 51" =

239,292 in3 ≈ 240,000 in

3

V = Abase × h = π r2 h =

π (24.8 cm)2 (45.4 cm) ≈

87,722 cm3 ≈ 87,700 cm

3

Some other common solids are the pyramid, cone and tetrahedron. All solids that have a flat base, a

similar cross-section and taper to a point have the same volume formula. Volume = (⅓) × Abase × height.

Note: The height is always measure perpendicular to the base.

Some Solids with a Similar Cross-Section

Pyramid

Cone

Tetrahedron

V = (⅓) x y h V = (⅓) π r2 h V = (⅓)(½)(x y)h = (1/6) x y h


Example 6 Find the volume of the following figures (a)

x = 425'

y = 415'

h = 510'

(b)

r = 1.12"

h = 4.25"

V = (⅓) x y h = 425' × 415' × 510' =

29,983,750 ft3 ≈ 30,000,000 ft

3

V = (⅓) π r2 h = (⅓) π (1.12")

2 (4.25")

≈ 16.748 in3 ≈ 16.7 in

3

The prior solids can all be made using multiple flat pieces. However, one

solid that cannot be formed using flat pieces is the sphere. The formulas for

volume and surface area of a sphere are not intuitive as in the case of prisms.

Surface Area = SA = 4π r

2 Volume = V = (4/3) π r

3

Example 7 Use a radius of the Earth (RE) of about 4,000 mi to estimate the Earth's surface area in sq-mi and its

volume in cu-mi.

SA = 4π r2 = 4π (4,000 mi)

2 ≈ 2.0 × 10

8 sq-mi

V = (4/3) π r3 = (4/3) π (4,000 mi)

3 ≈ 2.7 × 10

11 cu-mi

Practice 1.11.4 Volume of the following figures (a)

(b)

(c)

x = 36', y = 32', z = 28' x = 6.0', y = 4.0', z = 3.0' r = 12.00"

Answers

(a) 10,752 ft3 ≈ 11,000 ft

3 (b) ≈ 36 ft

3 (c) ≈ 7,238 in

3

1.11 Exercises Find the area and perimeter of the following figures.

1)

2)

3)

4)

5)

6)


7)

8)

9)

Find the perimeter as __ft __in.

10)

11)

12)

Find the area of these figures as sq-ft.

13)

14)

15)

Find the area of these figures as sq-in.

16)

17)

18)

Find the volume of these figures.

19) Rectangular

Prism

x = 48

y = 36

z = 55

20) Triangular

Prism

x = 7.8

y = 9.6

h = 12.5

21)

Triangular Prism

x = 52.8

y = 33.6

z = 205.5

22) Circular

Cylinder

r = 17.2

h = 32.6

23) Pyramid

x = 27.8

y = 29.6

h = 22.5

24) Tetrahedron

x = 68

y = 72

h = 54

25) Cone

r = 47.2

h = 32.6

26) Sphere

r = 2,400

27) Sphere

d = 182


1.12 Some Graphing Basics

Graphing uses some specific terminology with which you want to become familiar. Although numerous

types of graphs are in use in the world, in this section we focus on the two-dimensional rectangular

coordinate system which is commonly referred to as the Cartesian Coordinate System named after

French mathematician René Descartes (1596–1650). This section is meant to serve as a review or brief

introduction. A more in-depth discussion of graphing occurs in greater detail in Chapter 5.

1.12.1 Points in the Cartesian Coordinate System

In a two dimensional (2-D) rectangular coordinate system every unique point in the plane consists of a

corresponding unique pair of coordinates. When we list the coordinates we always give the horizontal

coordinate first and the vertical coordinate second. Since we usually use "x" to label the horizontal axis

and "y" to label the vertical axis, coordinates are typically referred to as (x, y)-points or ordered pairs.

The Cartesian Coordinate System Quadrants Standard Orientation

x-axis

y-a

xis

x

y(x, y)

origin

I II

III IV

x

y

When a graph is in standard orientation, towards the right is the positive direction for "x" while towards

the left is the negative direction for "x". Similarly, upward is the positive direction for "y" while

downward is the negative direction for "y". Although a graph is often in the standard orientation it is not

required.

Notice that point (1, 4) ≠ (4, 1). The order of the coordinates is

crucial, they cannot be interchanged, hence the phrase ordered pair.

When we plot a point, we get to the same location whether we start

with "x" or "y". Here we get to (4, 1) either by shifting 4 right then 1

up or by shifting 1 up then 4 right.

A single point can be designated by any ordered pair. It is

common to use letters such as (x, y) or (a, b) to indicate an

unknown point.

When there are multiple points using additional letters of the

alphabet can be limiting so we often use subscripts such as (x1, y1)

for the first point and, (x2, y2), (x3, y3), etc. for subsequent points.

Another common label is to use a single subscripted letter such as

P0, P1, P2, etc. to designate points.

Here (x1, y1) = (4, -4) and P4 = (-2, -3).


Practice 1.12.1

(a) Give the coordinates of the points A – J.

(b) Give the coordinates of the four corners of the rectangle.

(c) Give the coordinates of the three vertices of the triangle.

(d) Give the coordinates of the center of the circle.

(e) What area covers the region bounded by the origin, (0, 9) & (12, 0)?

A

B

C

DE

F GH

JI x

y

5 10

5

10

Answers

(a) A = (11, 11), B = (2, 9), C = (0, 7), D = (-3, 4), E = (6, 3), F = (-2, 0), G = (9, 0), H = (0, -1), I = -3, -2)

(b) (4, 5), (11,5), (11, 10), (4, 10) (c) (-2, 3), (13, -3), 6, 8) (d) (6, 9)

(e) These points form a triangle with base = 12 and height = 9. A = ½ b h = 54

1.12.2 Graphic Relationships

A graph can consist of individual points or we can connect the points in a line or a curve. We often use

the (x, y)-coordinate system for generating precise geometric diagrams. Many graphs (lines, circles,

etc.) illustrate a geometric relationship between the variables "x" and "y". We also use graphs to

represent a cause-effect relationship between two variables.

When a graph is used to illustrate a cause-effect relationship we call the "x-variable" the independent

variable and we call the "y-variable" the dependent variable. The terms independent and dependent

variable may seem confusing. If so, you may prefer to associate the "x-variable" with the

cause/input/stimulus and associate the "y-variable" with the effect/output/response in the relationship.

It should be noted that in many cases we prefer to use letters other than x and y for variables. We

commonly pick mnemonics (v for velocity, t for time, etc.) to represent our variables. In such cases, the

horizontal variable is always the independent variable (cause/input/stimulus).

1.12.3 Key Points for Interpreting x vs. y Graphic Relationships

Every graph illustrates a relationship between two variables. This relationship can be merely a

geometric shape such as a diagram, it can represent an algebraic relationship or it can represent a

narrative relationship. Although we shall further develop these concepts later on in the text for now it's

worthwhile to mention a few key ideas regarding the narrative interpretation of graphic relationships.

The graph of a line or a curve tells us a "story" about the relation between "x" & "y". Just as

philosophers say, "Every picture tells a story" every graph tells a story too.

Reading a Graph's Story

We normally read left to right and in mathematics we use that as an underlying rule too

when reading/describing a graph. Since we always describe a graph from left to right it is

not necessary to continually reiterate that "x" is getting bigger.

Remember, data is always plotted with "x" representing the cause/input/stimulus and "y"

as the effect/output/response. Since we invariably want to focus on results we primarily

describe the behavior/trend of the "y-variable" as we move left to right.


As we describe the behavior/trend of the "y-variable", we try to use phrases that have

every-day physical meaning in terms of the variables and their units. For instance

suppose the "y-variable" is labeled "v" and represents mph (velocity). Further, suppose

the graph is going up to the right. Instead of saying "y is going up" or "'v' is going up" it

would be better to say "the velocity is increasing", "the object is speeding up" or "the

object is accelerating". This puts the description into terms we can better relate to.

Points where a graph starts, stops, abruptly changes behavior, crosses an axis, intersects

another curve, instantaneously changes direction or curvature have special meaning and

are referred to as critical points. Behavior at critical points should be included in any

comprehensive description while the region between the critical points is usually not

mentioned in much detail. After all, when we describe our own lives, we focus on the

key events in our lives. In a graph, the key events correspond to the graph's critical

points.

Example 1 Briefly describe each graph

y+

x+

y+

x+

$

yrs

°F

min

y is increasing

y is going up

y is decreasing

y is going down

the cost is increasing

the cost is rising

the temperature is

decreasing

Example 2

Describe this graph which represents cost per credit hour during the last

15 years.

15 years ago, the cost/cr-hr was about $20. During the next 5 years the

cost increased slowly to about $30. Then a period of rapid increase

occurred and lasted for about 5 yrs. Near the end of the rapid increase the

cost approached $80. During the last 5 years the cost has continued to

increase (slowly) and is now about $100.

Cost per Credit Hr

5 yrs

$

10 15

50

100

Example 3

Describe this graph which represents the speed of an automobile.

This graph shows a car that started from rest (0 mph), and then sped up

(accelerated) quite rapidly for some minutes until it reached a high speed (a

maximum speed). Just after it reached this maximum speed it came to a

screeching halt (0 mph) almost immediately. Perhaps the driver slammed

on the brakes. It stopped briefly before taking off once again but this time

accelerating more slowly.

Smokey the Bandit

mph

min


Example 4 Describe this graph which represents someone's heart rate.

Here we see that the patient's heart rate began at 50 bpm, rose slowly

at first and then began to increase more rapidly until around 60 sec

when an obvious anomaly occurred consisting of a quick but short

drop in pulse. After that, the heart rate increased once more (even

more rapidly) until it peaked around 180 bpm at 80 sec. The patient's

heart rate then began to slow down, quickly at first then more

gradually This trend continued until the test ended at 180 sec.

bpm

180 sec

50

60 80

In the last example, the first downward spike, the subsequent upward spike and the maximum are all

critical points. If this was your heart, wouldn't you be most concerned about those points and the

interpretation of them in terms of your health? Hopefully you can see why we want to pay special

attention to the critical points.

Example 5

Describe this graph which represents the number of elk in The

Three Sister's Wilderness during the last 15 years.

Here we see that the initial elk population was about 80. It

increased slowly at first, then more rapidly before peaking at

about 250 after 5 years. This was followed by a rapid decline in

their population for the next 3 years until their population had

dropped to about 60 at year 8. After that the population slowly

recovered to about 90 elk when the study ended at year 15.

Three Sisters Elk Herd

Elk

yrs

80

5 8

Example 6 Describe this graph which represents the number of acres that have burned relative to the fire suppression

budget for that year.

Here we see that if no money is spent we expect about 800,000

acres to burn. However, as we begin to spend money on fire

suppression we initially obtain a large decrease in acres burnt.

And as we spend more money on suppression the acreage

burnt continues to decline. However, the effectiveness of the

suppression goes down. Above $50M there is a less rapid

decrease and after about $100M there is very little success in

the lowering the acreage burnt. That is, although fewer acres

burn, the decrease in burnt acreage is smaller and smaller for

larger and larger expenditures. It appears that no matter how

much we eventually spend we'll always have at least 200,000

acres burn.

Fire Suppression Budget acres

burnt

$-spent$50M

200,000 ac

If you had to budget for fire suppression, how much would you allocate in your budget; $25M, $50M,

$100M? There is no right or wrong answer here. But, we can use the above graph to help us make an

informed decision.

A great deal of effort in a wide variety of professions (from EMT's reading a cardiogram to Fish &

Game trying to manage an Elk herd) goes in to collecting data, plotting the data, interpreting the


resulting graph then making decisions based on that interpretation. We study this in greater detail in

later chapters.

Remember, we always describe the graph from left to right, we always describe the behavior of the

graph in terms of the action of the "y-variable" and we try to expressly include any critical points.

1.12 Graphing Exercises

1) Give the coordinates of the points A, B, C.

2) Give the coordinates of the points D, E, F.

3) Give the coordinates of the points G, H, I.

4) Plot and label P = (7, 9), Q = (10, 2), R = (5, 0)

5) Plot and label S = (0, 8), T = (-2, 6), U = (-3, -4)

6) Plot and label P1 = (-4, 8), P2 = (-4, 0), P3 = (10, 0)

10

5

5 10

A B C

D

E

F G

H

I

7) Give the coordinates of the corners of the rectangle.

8) Give the coordinates of the vertices of the triangle.

9) Give the coordinates of the center of the circle.

What is its radius?

10) (-2, -2), (-2, 12) and (8, 12) form 3 corners of a rectangle.

What are the coordinates of the 4th corner?

11) (2, -3), (2, 7) and (11, 7) form 3 corners of a rectangle.

What are the coordinates of the 4th corner?

x

y

5 10

5

10

12) Oregon Fish and Game (OFW) recently conducted a study of the Black Butte elk herd. Answer the

following questions based on their graph of the Elk Population vs. Year.

(a) When did the study start?

(b) How long did the study last?

(c) For how many years was the elk population in a state of

decline during the study period?

(d) What was the minimum population of the elk herd

during the study period?

(e) What is the elk herd's current trend? How does that

compare to 1985?

Black Butte Herd 1985-2003

Elk

50

1990 19951985 2000

100

150

Yr

(f) Based on your answer to (e), what would be a reasonable prediction for the elk herd over the next 5

years 2003-2008?

13) Frank wants to make first tracks on the mountain so he gets up early, gets dressed and quickly drives off

toward Mt. Bachelor. It's slow through town but once on Century Drive he drives very fast until he reaches

a long string of cars moving quite slowly on the icy road. Frank slows to a crawl but the slow pace drives

him nuts and so he begins passing the long string of slow moving cars all together. As he passes the 20th


and last car in the long string his radar detector goes off. Busted! He quickly pulls in front of the last car

and slows down but the damage is done. He won't have first tracks now.... Answer the following questions

based on the graph below.

(a) At what speed does Frank drive through town?

(b) How long does it take him to drive through town?

(c) How fast does he initially drive outside of town?

(d) Put a 'T' on the graph corresponding to where Frank begins

tailgating.

(e) For how long does he tailgate before beginning to pass?

(f) What is his fastest speed while passing?

(g) Put an 'R' where the radar detector must have gone off.

Frank's Speeding Ticket

(h) How long did it take to get the ticket?

(i) At what speed does he drive after receiving the ticket?

14) Judy is training for her 5th marathon. She begins her workout with some warm-up exercises then does a

short cool down before heading out on a run. At the end of the run she drops to a jog. Judy has the run and

jog each last the same amount of time. After the jog she does a longer cool down. Answer the following

questions based on the graph below which gives Judy's heart rate over time.

(a) What was Judy's heart rate at the start of the

workout?

(b) How long did her warm-up exercises last?

(c) What average heart rate did she maintain during her

warm-up?

(d) Place a "C" at her first cool-down period.

(e) Place an "R" where she begins her run.

Judy's Heart Rate

(f) Place a "J" where she begins to jog.

(g) What was her maximum heart rate during her run?

(h) What was her average heart rate during her jog?

(i) How long did the run plus jog last?

(j) Place an "F" where she finishes jogging.

15) This graph represents the 2004 cumulative rainfall for Riley.

(a) Approximately how much precipitation fell in Jan?

(b) Approximately how much precipitation fell in Feb?

(c) Approximately how much precipitation fell in April?

(d) What was the driest month?

(e) What was the wettest month?

(f) What was the total precipitation for Riley From Jan 1 to Jul 1?

(g) What was the average monthly rainfall?

Riley's Precipitation

inch

es r

ain

fall

10

20

Jan

1

Feb

1

Mar

1

Ap

r 1

May

1

Jun

1

Jul

1


16a) This graph represents the 2002 monthly rainfall for Sun City.

(a) About how much rain fell in May?

(b) Approximately how much rain fell in

October?

(c) Was April relatively wet or dry?

(d) What were the driest months?

(e) What was the total rainfall for June,

July and August combined?.

rainfall

0.0 in

1.0 in

2.0 in

3.0 in

4.0 in

5.0 in

6.0 in

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

rainfall

(f) What was the total yearly rainfall for Sun City? (g) What was the average monthly rainfall?

16b) This graph represents the 2003 cumulative rainfall for Sun City.

(a) There was a huge winter storm.

Place a "W" when that must

have occurred

(b) Approximately how much rain

fell in March?

(c) What was the driest period?

(d) What were the wettest weeks?

(e) There was a large spring storm.

Place a "S" when it must have

occurred.

cummulative rainfall

0.0 in2.0 in4.0 in6.0 in8.0 in

10.0 in12.0 in14.0 in16.0 in

1/1

/2003

1/8

/03

1/1

5/0

3

1/2

2/0

3

1/2

9/0

3

2/5

/03

2/1

2/0

3

2/1

9/0

3

2/2

6/0

3

3/5

/03

3/1

2/0

3

3/1

9/0

3

3/2

6/0

3

4/2

/03

4/9

/03

Week of

(f) What was the total rainfall for Sun City for 1/1/03 - 4/9/03?

(g) What was the average monthly rainfall?

(h) How much rain fell in January?

Points and Geometry on a Rectangular Coordinate System

17) Plot these points and connect them to make a square. (2,3), (2,8), (-3,8), (-3,3)

18) Plot these points and connect them to make a square. (5,-2), (2,1), (5,4), (8,1)

19) Plot these points and connect them to make a square. (-3,3), (0,4), (1,1), (-2,0)

20) How do you know that the four points in problem 17, 18 and 19 actually do form a square?

i.e. how do you know that all the sides are the same length, and that all the angles are right angles?

21) Plot these points and connect them to make a rectangle. (-5,-8), (7,-2), (4,4), (-8,-2)

22) Plot these points: (2,-5), (2,-2), (5,-2). Find a fourth point (x, y) to make a square.

23) Plot these points: (-1,6), (-5,2), (-1,-2). Find a fourth point (x, y) to make a square.

24) Plot these points: (1,-2), (-1,1), (2,3). Find a fourth point (x, y) to make a square.

25) Plot these points: (0,6), (-4,5), (-2,-3). Find a fourth point (x, y) to make a rectangle.


26) A quadrilateral is a four-sided polygon. Quadrilaterals can be identified by connecting their vertices.

Sketch the five quadrilaterals defined by the sets of coordinates given here. Then name the quadrilateral

(parallelogram, square, trapezoid or rectangle).

a) (2,3) (2,8) (10, 3) (10,8) b) (-2,-3) (-2,3) (4,3) (4, -3) c) (-1,-1) (3,3) (3,-5) (7, -1)

d) (0, -6) (0,8) (-4, -4) (-4,6) e) (2,2) (3,7) (11,6) (12, 11)

27) For each of the following four problems, three vertices of a quadrilateral are given. Determine the

coordinates of a fourth vertex.

a) Square: (-1,-1) (-2,5) (4,6) (?, ?) b) Rectangle: (0,0) (2,-3) (-6,-4) (?, ?)

c) Parallelogram: (4,2) (6,8) (12,11) (?, ?) d) Trapezoid: (-6,-3) (-3,6) (6,6) (?, ?)

28) Plot the points (2,-3) and (-2,3). These are two vertices of a quadrilateral. Locate and give the coordinates

of two other vertices so the quadrilateral is…

(a) a square (b) a square different than (a) (c) a rectangle that is not a square

(d) a parallelogram that is not a rectangle (e) a parallelogram other than (b), (c), or (d)

(f) a trapezoid. There are multiple answers to this problem.

29) Plot the points (5,2), (2,6), and (10,9). These are three vertices of a parallelogram.

(a) Locate and give the coordinates of a fourth vertex.

(b) Locate and give the coordinates of another point which could be the fourth vertex.

(c) How many different points could be a fourth vertex? Give the coordinates of any other vertex you can

find.


Chapter 1 Review Exercises

1) (a) T F 93 = 999 (i) T F 2 + 3/4 =

2 + 3

4

(b) T F 6 sq-ft = 6 ft2 (j) T F 4

2 = 44

(c) T F 3(4 + 5) = 3[4 + 5] (k) T F 2{ 3 + 4[4 + (2)(9)] } = (2)(91)

(d) T F 4/5 = 80% (l) T F 3.14 < < 3.15

(e) T F 3.5 > 7/2 (m) T F 0.001 > 0.0002

(f) T F Q2 = Q·Q (n) T F 54 = (5)(5)(5)(5)

(g) T F 3/5 = 3

5 (o) T F 3/5 = 0.6

(h) T F 8 – (2)(4)

(2)(3) – 6 = 0 (p) T F =

2) 2,345.6789 has a 4 in the place and an 8 in the place.

3) Write numerically, seventy five million. Include commas.

4) Write algebraically, the sum of the squares of A and B.


66 = (b)

14

112 =




3 = (b)

933

5 =

9) Convert to an improper fraction: (c) 7 3

5 = (d) 4

2

3 =

Compute and give your answer as a fraction:

10) (a) 2

3 +

5

3 = (b)

7

15 +

7

12 = (c) 7

3

5 – 2

1

3 =

11) (a) 2

3 ·

5

8 = (b)

2

3 ÷

5

12 = (c) 1

3

5 · 2

1

3 =

Simplify the calculation to a single value:

12) (a) (3 + 4)10 – 5 = (b) 8·32 + 7 = (c) 10 ÷ 12 ÷

5

6 =

13) (a) 4 · 5 – 3 = (b) 52 – 3

2 = (c) (3 + 2)10 – 5 =

14) (a) 6 + 9 + 10

2 + 3 + 5 = (b)

(6 – 4)(5 – 3)

6 + 4 – 2 · 5 = (c)

6 · 15

25 · 9 =

15) (a) -22 = (b) 3

2 + 4

2 = (c) 8(6 – 9)10 – 5 =

16) (a) 16 – 3 · 5 + 5 = (b) 3 · 10 – 10 · 2 = (c) (3 · 5)2 + 3 · 5

2 =


Use a calculator to obtain a single value accurate to 2 decimal places:

17) (a) 1000 = (b) 8 · 3

2

7 + 4 = (c)

16.5 · 12.2

14.8 · 7.5 =

18) (a) 1 + 2.32 = (b)

2.4 + 3.5

2.42 + 3.5

2 = (c) 12.2 − 69.3

4.8 · 1.5 =

Use substitution to evaluate accurate to 2 decimal places:

19) V = 3P + 4Q P = 2.6, Q = 1.5 V =

20) T = D

1 + P2 D = 9.8, P = 0.15 T =

21) V = 1

2 (R

2 – r

2) H R = 32.5, r = 3.5, H = 6.9 V =

22) H = a2 + b

2 a = 24, b = 7 H =

23) A = b1 + b2

2 · h b1 = 7.2, b2 = 6.8, h = 12.5 A =

24) s = (x̄ – xi)

2

n – 1 x̄ = 4.5, xi = 3.8, n = 65 s =


(a) 3

8 =

x

24 (b)

3

8 =

x

12 (c)

3

x =

5

12

26) Round to…

(a) hundredths: 3.092165 (b) tens: 253.092165 (c) thousandths: 43,982.0349821

27) Compute and round accordingly. 2.69" + 175" – 24.25 + 0.005" ≈

28) Compute and round accordingly. 541' x 1,292' ≈

29) Use a calculator to evaluate and round to the appropriate number of significant digits:

(a) (6.42)(167)

5.3 ≈ (b)

3.4

53.5 · 1.60 ≈ (c) 530·1.602 ≈

(d) 3.14·5,287 ≈ (e) 2500· 1.2 ≈ (f) 3.14159·5,287 ≈

30) What accuracy is required in measuring the base & height dimensions in order to compute the area of a

triangle (which is approximately 50' x 30') to the nearest sq-ft?

31) In algebraic form write:

(a) the quotient of input to output using in & out.

(b) the ratio of lead to zinc using L & Z.

(c) the product of length and width using L & W.

(d) the average of three scores using T1, T2, T3.

(e) the difference of P and Q

(f) the difference between the squares of a and b.


32) List the increments in this ruler.

(not drawn to actual size) 1" 2"

33) List the increments in this metric ruler.

(not drawn to actual size) 21

1.0 cm

34) A rectangular garden is 50' x 30'. What is the area? What is the Perimeter? 50'

30'

35) What are the area and perimeter of the triangle? 5"12"

13"

36) What are the area and perimeter of the triangle?

42' 28' 34' 11'

37) What are the circumference and area of a 10.0" diameter circle?

10"

38) Circle the trapezoid(s):

39) Plot the points: A=(2, -3),B=(-4,-5), C=(4,-6), D=(-2,8), E=(-6,5),

F=(4,4)

40) Plot these points: (-1,6), (-5,2), (-1,-2). Find a fourth (x, y)- point

to make a square. (x, y) =

41) Find the missing dimensions

B = 43

8", a = 1

1

2", c =

7

8"

H = 31

4", d = 1

1

2", e =

5

8"

a

b

cdeH

B

h

42) Find the missing dimensions

H = 17

8", B = 2

1

4", a =

1

4", d =

3

8"


Answers-

1) F,T,T,T,F,F,T,F,F,F,T,T,T,T,T,F

2) tens, thousandths

3) 75,000,000

4) A2 + B

2

5) 2/11, 1/8

6) 1/4, 2/5

7) 153/25, 33/100

8) 24 1/3, 186 3/5

9) 38/5, 14/3

10) 7/3, 21/20, 79/15

11) 5/12, 8/5, 56/15

12) 65, 79, 1

13) 17, 16, 45

14) 2 ½, Ø, 2/5

15) –4, 5, -245

16) 6, 10, 300

17) 31.62, 6.55, 1.81

18) 2.51, 0.33, -7.93

19) 13.80

20) 9.69

21) 11,315.39

22) 25.00

23) 87.50

24) 0.01

25) 9, 9/2, 36/5

26) 3.09, 250, 43,982.035

27) 153

28) 699,000

29) 200, 0.01, 850, 16,600, 3,000, 16,610

30) 4 significant digits, i.e. 50.xx' x 30.xx', each

must be accurate to nearest hundredth

31) in

out ,

L

Z , L·W,

T1+T2+T3

3 , P−Q, a

2−b

2

32) 5/16, 5/8, 1 1/32, 1 15/32, 1 15/16

33) 3mm, 0.75cm, 1.05cm, 1.4cm, 1.65cm

34) A=1500 sq-ft, P=160 ft

35) A=30 in2, P=30 in

36) A=231 sq-ft, P=104 ft

37) C=10π"≈31.4", A=25π in2 ≈78.5 in

2

38) *

39) *

40) *

41) b=2", h=2 ⅜"

42) b=1", c=3/4"

38)

39) y

x

40) y

x


Chapter 1 Test

1) (a) T F 43 = 444 (f) T F 2 + (3/4) =

2 + 3

4

(b) T F P2 = P·2 (g) T F 1/3 = 0.33

(c) T F 3(4 + 5) = 3{4 + 5} (h) T F 10

0 = 0

(d) T F 13

20 = 65% (i) T F

8 – (2)(4)

(2)(3) – 6 = 0

(e) T F 3 < < 3.14 (j) T F =


54 = (b)

32

112 =



3 = (b)

47

5 =

5) Convert to an improper fraction: (c) 5 1

8 = (d) 2 3

5 =

6) Compute and give your answer as a reduced fraction:

(a) 2

15 +

5

12 = (b)

5

8 ·

12

35 = (c) 1

2

3 · 4 5

8 =

Simplify the calculation to a single value:

7) (a) (3 − 7)10 – 5 = (b) 7·32 + 8 = (c) 12 ÷ 16 ÷

3

4 =

8) (a) -4 · 6 – 3 = (b) 52 – 6

2 = (c) (2 − 3)10 – 5 =

9) (a) 26 + 94

13 + 17 = (b)

(7 – 11)(5 – 13)

-2 · 5 = (c) -3

2 =

10) Use a calculator to obtain a single value accurate to 2 decimal places:

(a) 500 = (b) 7.2 − 9.3

4.8 · 1.5 = (c)

16.5 − 12.2

14.8 · 7.5 =

Use substitution to evaluate accurate to 2 decimal places:

11) V = 3P + 4Q P = 6.97, Q = 8.42 V =

12) T = D

1 + P2 D = 11.42, P = 0.13 T =

13) H = a2 + b2 a = 37, b = 19 H =

14) A = b1 + b2

2 · h b1 = 17.24, b2 = 12.85, h = 12.5 A =


(a) 13

16 =

x

128 (b)

12.5

x =

5

12 (c) 0.515625 =

x

64


16) What accuracy is required in measuring the base & height dimensions in order to compute the area of a

triangle (which is approximately 250' x 90') to the nearest 10 sq-ft?

17) List the increments in this ruler. (not drawn to actual size)

1" 2"

18) List the increments in this metric ruler. (not drawn to actual size) 21

1.0 cm

19) Plot the points:

(5, -8), (-9,-5), (10,-6), (0,-8), (-6,0)

y

x

20) H = 4 3/8", h = 3 5/16", a = 1 7/16", c = 1 9/16"

(a) Find b (b) Find B

ab

d

c

ef

B

Hg

h


Chapter 1 Project- Solar Panel Alignment

To be mathematically literate it is imperative that you be able to effectively communicate mathematical concepts

in a written fashion using correct mathematical notation. It is important that you be able to understand diagrams,

create diagrams and use diagrams to generate mathematical relationships. This assignment is one step in helping

you become both more mathematically literate and review the mathematics of angles.

Aligning

Solar

Panels Sun

figure 1

N

S

Sun23.5°

44.0°

Equator

Position of Sun

and Earth

on Winter Solstice

44° latitude

?

figure 2

Solar energy is an ever more important component in our energy needs. In order for solar panels to be most

effective they would need to be angled so that the panel's face is perpendicular to the sun. See figure 1.

Unfortunately the only way this can be accomplished is for the panel to both rotate throughout the day (follow the

sun) and change its inclination (angle off the horizontal) throughout the year to accommodate the tilt of the Earth

as it orbits around the sun. A simple/cheap alternative is to fix the panel at one 'average' angle.

Assignment What angle (off the horizontal) would a solar panel need to have so that a panel set on the ground would be

perpendicular to the sun (at the sun's high point) at the Winter Equinox?

Write a paper (one side of one page maximum length) that outlines the above problem and then answers the

question. Your paper must follow the guidelines given below. It will be graded on presentation, completeness,

accuracy, punctuality and approach to the problem. It will be graded using the attached rubric which should be

attached to your paper when submitted.

Guidelines

Your paper must be typed or neatly handwritten

Your paper (with diagram) must neatly fit entirely on one side of one page.

Attach (staple) this page but all the pertinent information including diagrams must be on your page.

There should be a Title and 3 distinct sections: Introduction, Solution, and Conclusion.

Your Introduction should include some human interest to motivate the purpose of aligning solar panels.

Your Introduction should include a clear problem statement (your paper's purpose) in your own words so

that someone not familiar with this assignment would understand the purpose of the paper.

The Solution must clearly show/describe the calculations in a step by step process.

The paper should assume the ground is flat (perpendicular to the radius of the Earth).

The Earth's axis of rotation is tilted 23.5° from the plane of its orbit. This is a known fact. You do not

need to derive this angle.

Assume the house is located at about 44° N latitude. (Bend's latitude)

Your paper must contain at least one well placed diagram that is appropriately labeled and enhances your

paper.


Solar Assignment Grading Rubric NAME

OVERALL FORMAT- Layout/Organization/Presentation

Typed with clearly readable font or neatly handwritten. Uses title and other clarifying headings. Layout and

information organization/presentation flows for easy readability.

Includes Title, Includes all Headings Appropriate / Clear Notation

Section Breaks Apparent Appropriate Layout of Multiple Steps

Appropriate use of White Space Spelling / Grammar

Easy on the Eyes / Overall Readability Appropriate for Audience

Clear Well Placed Diagram(s) w/ Credit Fits Nicely on One Page

Solution is Readily Apparent Rubric is Stapled to Report

INTRODUCTION

Clear, easy to read. Provides motivation, includes clear problem statement and flows naturally into the General

Procedure. Makes the reader want to continue reading. Diagrams enhance Introduction.

Includes Heading Clear Problem Statement (purpose of paper)

Human Interest Included Human Interest Appealing

SOLUTION

Processes/Strategies/Calculations used follow paper's General Procedure. They are easy to follow, accurate,

complete and lead to a correct solution. Charts/diagrams enhance the paper.

Includes Heading Diagram(s) Enhance Explanation

Includes Justification of Solution Method Explanation/Justification is Easy to Follow

Clearly Marked Specific Steps Appropriate Calculation Detail

Result(s) Correct Result(s) Easily Identified

CONCLUSION/SUMMARY

Conclusion/Summary is easy to ready and is consistent with the Introduction.

Includes Heading Ties in with Introduction


sun 44.0°

23.5°

sola

r pa

nel

Equator

44° N latitude

N

S

flat to Earth

?

Winter Solstice

Alignment

23.5°

a

b

c d

Some Hints

We want to find angle 'd'

There are many ways to approach this problem! One approach:

Use 180° in a triangle to find 'a'.

Use the parallel nature of the sun's rays to find 'b'.

Use 180° in a straight line to find 'd' which is the angle we want.


SAMPLE SAMPLE SAMPLE

The Earth's Fate is in Your Hands

Tit

le

INTRODUCTION

Global Warming is threatening Earth. Mathman can help reduce carbon production if he

can make solar energy more efficient. Fortunately this is easy to do by just understanding

the angles of the Sun …..

Intr

od

uct

ion

w

/ hum

an i

nte

rest

and p

roble

m s

tate

men

t

The Problem …. This paper shows

how to determine the optimum

angle for a solar panel at the

Winter Solstice when the solar

panel is located in Bend, OR.

N

S

Sun23.5°

44.0°

Equator

Position of Sun

and Earth

on Winter Solstice

44° latitude

?

SOLUTION

Find angles of the Sun pertinent to solar panel orientation requires some basic geometry

…..

Step 1

Step 2

…

Solar Panel Angle

So

luti

on

answ

er e

asi

ly i

den

tifi

ed

SUMMARY / CONCLUSION

With this procedure we can make solar energy more efficient. .….. The Earth is now better

off!

Co

ncl

usi

on

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