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Title: Arithmétic exercices !!! for only 50 $ !!!!!
Description: you need arithmétic exercices ? defintions ? here you are !!!!!
Description: you need arithmétic exercices ? defintions ? here you are !!!!!
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ARITHMETIC:
A Textbook for Math 01
3rd edition (2012)
Anthony Weaver
Department of Mathematics and Computer Science
Bronx Community College
Page 2
3rd Edition
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Thanks to the following colleagues for various combinations of proof-reading, technical help, improvements in pedagogy and/or exposition:
Nikos Apostolakis, Luis Fernandez, Marie Hercule, Uma Iyer, Toni Kasper, Alexander Kheyfits,
Roman Kossak, George Leibman, Emanuel Paki, Maria Psarelli, Philipp Rothmaler, Camilo Sanabria
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– A
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June, 2012
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0
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1
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1
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1 Commutativity, Associativity, Identity
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1
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1
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3 Exercises
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2 Subtracting Whole Numbers
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2
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1
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2 Multi-digit subtractions
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2
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1
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4 Borrowing
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2
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1
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1
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1 Commutativity, Associativity, Identity, the Zero Property
1
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2 Multi-digit multiplications
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3
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1
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1
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1 Squares and Cubes
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4
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1
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3 Square Roots
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4
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1
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1
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1 Quotient and Remainder
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5
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1
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3 Exercises
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6 Order of operations
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6
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1
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1 Exercises
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8 Perimeter, Area and the Pythagorean Theorem
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2 Fractions and Mixed Numbers
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2
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1 Zero as Numerator and Denominator
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2
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1 Converting an improper fraction into a mixed or whole number
2
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2 Exercises
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3
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2
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4 Exercises
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4 Multiplication of Fractions
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4
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2
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2
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1 Cancellation and Lowest Terms
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5
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2
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2
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1 Exercises
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6
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2
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3 Exercises
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6
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2
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5 Exercises
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7 Pre-cancelling when Multiplying Fractions
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7
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2
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2
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1 Exercises
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8
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2
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3 The LCM
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8
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2
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5 The LCD
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8
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2
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2
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1 Exercises
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10 Division of Fractions
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10
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2
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2 Exercises
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10
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2
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4 Exercises
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11 Mixed Numbers and Mixed Units
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11
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2
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2 Exercises
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11
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2
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4 Exercises
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12 Combined operations with fractions and mixed numbers
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12
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3 Decimals and Percents
3
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3
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1 Exercises
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2 Significant and Insignificant
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3
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1 Exercises
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91
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3
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3
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1 Exercises
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5 Adding and Subtracting Decimals
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5
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3
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6
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3
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3
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1 Exercises
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8 Division of a decimal by a whole number
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8
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3
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3
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1 Exercises
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10 Percents, Conversions
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10
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3
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3
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1 Exercises
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95
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101
101
102
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108
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4 Ratio and Proportion
4
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4
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1 Exercises
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2 Proportions
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2
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2
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4
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3 Exercises
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3 Percent problems
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3
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4
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4
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1 Exercises
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5 Similar triangles
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5
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115
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118
119
122
122
124
125
126
127
131
5 Signed Numbers
5
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5
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1 Exercises
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1
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5
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3 Exercises
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1
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5
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5 Exercises
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2 Subtracting signed numbers
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2
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5
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3
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5
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5
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1 Division and 0
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4
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5
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5
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1 Exercises
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135
135
139
140
140
141
142
143
145
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149
150
152
153
153
155
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6
5
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8
5
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6
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Evaluating Expressions
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7
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Using Formulae
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8
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Linear Equations in One Variable
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9
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5
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2 Exercises
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156
158
158
161
162
164
164
165
169
Chapter 1
Whole Numbers
The natural numbers are the counting numbers: 1, 2, 3, 4, 5, 6,
...
In order to write numbers efficiently, and for other reasons, we also need the
number 0
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Taken
together, all these numbers are called the integers
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There are numbers between any two integers on the number line
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Some locations represent fractions such as
one-half (between 0 and 1) or four-thirds (between 1 and 2)
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(π is located between 3 and 4 and expresses the ratio of
the circumference to the diameter of any circle
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We need only ten symbols to write any whole number
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We write larger whole numbers using a place-value system
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Example 1
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So if a number has five digits,
the fifth-from-right place indicates how many ten-thousands the number contains
...
)
9
1
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1
Exercises
1
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209 stands for
3
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21045 stands for
1
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We assume you know the sums of
single-digit numbers
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Example 2
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For example, the number in
the row labelled 3 and the column labelled 4 is the sum 3 + 4 = 7
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1: The digit addition table
1
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Why is the second line from the top identical to the top line?
3
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1
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Page 10
In other words, the order in which two numbers are added does not effect the sum
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The last two questions lead us to an important property of 0, namely, for any number x,
x + 0 = x = 0 + x
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Because of
this property, 0 is called the additive identity
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This property of addition is called associativity
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Find the sum of 2, 3, and 5 by associating in two different ways
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Associating 2 and 3, we calculate
(2 + 3) + 5 = 5 + 5 = 10
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The sum is the same in both cases
...
1
...
Then we add
the digits in each place to obtain the sum
...
Find the sum of 25 and 134
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We line up the numbers vertically so that the 5 in the ones place of 25 is over the 4 in the
ones place of 134
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So there is a “ones”
column, a “tens” column to the left of it, and a “hundreds” column to the left of that:
25
134
Then we draw a line and add the digits in each column to get the sum:
25
+ 134
159
Page 11
Sometimes we need to “carry” a digit from one place to the next higher place
...
But 15 has two digits: it stands for 1 ten
+ 5 ones
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So we have:
Example 5
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Put down the 5 in the ones place:
38
+ 47
5
and carry the 1 to the top of the tens column:
{1}
38
+ 47
5
and finish the job by adding up the tens column, including the carried one:
{1}
38
+ 47
85
The sum is 85
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1
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3
Exercises
Find the sums, carrying where necessary
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26
+
55
Page 12
2
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32
45
+
64
4
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909
777
+ 6964
6
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9
999
902
+ 9502
8
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32000
9844
902
+ 4503
10
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2
Subtracting Whole Numbers
Another way to say that 5 + 2 = 7 is to say that 5 = 7 − 2
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” The operation of taking away one number from another,
or finding their difference, is called subtraction
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We assume you remember the differences of single-digit numbers
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Example 6
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Here’s a start: the number
in the row labelled 7 and the column labelled 2 is the difference 7 − 2 = 5
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(Squares that have an asterisk (∗) will be filled
in later with negative numbers
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2: The digit subtraction table
1
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1
Commutativity, Associativity, Identity
When we study negative numbers, we will see that subtraction is not commutative
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Example 7
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Solution
...
Until we establish an order of operations, we will avoid examples like this!
It is true that
x −0=x
for any number x
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To make sense of 0 − x, we will need negative numbers
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2
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The latter number is called the
subtrahend, and the former, the minuend
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To set up the subtraction, we line the numbers up vertically, with the minuend over the subtrahend,
and the ones places lined up on the right
...
Find the difference of 196 and 43
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The subtrahend is 43 (the smaller number), so it goes on the bottom
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196
43
Then we draw a line and subtract the digits in each column, starting with the ones column and
working right to left, to get the difference:
Page 15
196
− 43
153
1
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3
Checking Subtractions
Subtraction is the “opposite” of addition, so any subtraction problem can be restated in terms of
addition
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In general, if subtraction has been performed correctly, adding the
difference to the subtrahend returns the minuend
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Example 9
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Adding the (supposed) difference to the subtrahend, we get
33
+ 51
84
which is not equal to the minuend (94)
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We leave it to you to fix
it!
1
...
4
Borrowing
Sometimes, when subtracting, we need to “borrow” a digit from a higher place and add its equivalent
to a lower place
...
Find the difference:
85
− 46
Solution
...
Instead
we remove or ”borrow” 1 ten from the tens place of the minuend, and convert it into 10 ones, which
we add to the ones in the ones place of the minuend
...
It looks funny (as if 15 were a digit), but it doesn’t change the value of
the minuend, which is still 7 × 10 + 15 = 85
...
We can check this by verifying that the difference + the subtrahend = the original
minuend:
39
+ 46
85
Sometimes you have to go more than one place to the left to borrow successfully
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Example 11
...
In the ones column we can’t take 9 from 7, so we need to borrow from a higher place
...
But we can borrow from the hundreds place
...
The minuend is now represented as
{1}{9}{17}, standing for 1 hundred + 9 tens + 17 ones, (as if 17 were a digit)
...
To check, we verify that the difference + the subtrahend = the original minuend:
138
+ 69
207
(You may have noticed that the carrying you do in the addition check simply reverses the borrowing
done in the original subtraction!)
1
...
5
Exercises
Find the differences, borrowing where necessary
...
1
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275
− 181
3
...
500
− 191
5
...
1500
− 1191
7
...
50000
− 4999
9
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6389
− 999
11
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9001010
−
1111111
Page 19
1
...
When we say “4 times 3 equals 12,” we can think of it
as starting at 0 and adding 3 four times over:
0 + 3 + 3 + 3 + 3 = 12
...
Using the symbol × for multiplication,
we write
3 + 3 + 3 + 3 = 4 × 3 = 12
...
Instead of the ×
symbol, we often use a central dot (·) to indicate a product
...
Example 12
...
For example, the number in the row labelled 7
and the column labelled 5 is the product 7 · 5 = 35
...
3: Multiplication table
1
...
Why does the second row from the top contain only 0’s?
3
...
Is multiplication commutative? How can you tell from the table?
Page 20
1
...
1
Commutativity, Associativity, Identity, the Zero Property
An examination of the multiplication table leads us to an important property of 0, namely, when any
number, N, is multiplied by 0, the product is 0:
0 · N = N · 0 = 0
...
For this reason, 1 is called the multiplicative identity
...
Example 13
...
On the left, we have piled
up 3 rows of 4 squares (3 × 4); on the right, we have piled up 4 rows of 3 squares (4 × 3)
...
Figure 1
...
Example 14
...
We could first associate 3 and
4, getting
(3 × 4) × 5 = 12 × 5 = 60,
or we could first associate 4 and 5, getting
3 × (4 × 5) = 3 × 20 = 60
...
1
...
2
Multi-digit multiplications
To multiply numbers when one of them has more than one digit, we need to distinguish the number
“being multiplied” from the number which is “doing the multiplying
...
It really makes no difference which number is called the
multiplier and which the multiplicand (because multiplication is commutative!)
...
To set up the multiplication, we line the numbers up vertically, with the multiplicand over the
multiplier, and the ones places lined up on the right
...
To multiply 232 by 3, we write
232
×
3
We multiply place-by-place, putting the products in the appropriate column
...
For
example, 4 × 5 = 20, a number with two digits
...
Example 16
...
Solution
...
This gives us 10 hundreds, or 1 thousand
...
{2}
251
×
4
1004
The product of 251 and 4 is 1004
...
We get partial
products (one for each digit of the multiplier) which are added to yield the total product
...
Consider the product
24
×
32
Since the multiplier stands for 3 tens + 2 ones, we can split the product into two partial products
24
×
2
48
and
24
×
3
Notice that in the second partial product the multiplier is in the tens column
...
The second partial product is obtained by simply putting down a 0 in
the ones place and shifting the digit products one place to the left:
24
×
3
720
(Notice that we put down 2 and carried 1 when we performed the digit product 3 × 4 = 12
...
We can write the whole procedure
compactly by aligning the two partial products vertically
24
×
32
48
720
Page 23
and then performing the addition
24
×
32
48
+
720
768
Here’s another example
...
Find the product of 29 and 135
...
We choose 29 as the multiplier since it has the fewest digits
...
Next we use the 1-digit multiplier 2 (standing for 2 tens) to obtain the second partial
product, shifted left by putting a 0 in the ones place
{1}
135
×
29
1215
2700
(What carry did we perform?) Finally, we add the partial products to obtain the (total) product
135
×
+
29
1215
2700
3915
Note that the whole procedure is compactly recorded in the last step, which is all that you need to write
down
...
3
...
1
...
83
×
5
3
...
3008
×
9
5
...
83
×
56
Page 25
7
...
308
×
109
9
...
837
×
1
...
The factor of 1 is understood and usually omitted
...
In the expression 34 , 3 is called the base, and 4 the exponent (or power)
...
The 5th power of 2, or 25 , is the product
2 × 2 × 2 × 2 × 2 = 32
...
We make the following definition in the cases where the exponent is 0
...
(00 is undefined
...
(You’ll see another justification when you study algebra
...
, with 0 base and nonzero exponent, make
perfect sense (e
...
, 03 = 0 × 0 × 0 = 0)
...
170 = 1
...
4
...
01 = 0
...
Squares and Cubes
Certain powers are so familiar that they have special names
...
Thus 52 is read “5 squared,” and 73 is read as “7 cubed
...
8)
...
This means that the square contains x 2 small squares, each one unit on a side
...
=
Similarly, the volume of a cube, y units on a side, is y 3 cubic units
...
For example, the volume of an ice cube that measures
2 cm (centimeters) on a side is 23 = 8 cubic centimeters
...
4
...
Rewrite using an exponent: 8 × 8 × 8 × 8
2
...
Evaluate 25
4
...
Evaluate 07
6
...
Evaluate 102
Page 27
8
...
Complete the following table of squares:
02
12
22
32
42
52
=0
=
=
=
=
=
62 =
72 = 49
82 =
92 =
102 =
112 =
122
132
142
152
162
172
182
192
202
302
402
502
=
=
=
=
=
=
=
=
=
=
=
=
10
...
4
...
For example, 22 = 4, so 2 is the square root of 4, and we write
√
4 = 2
...
Actually, every positive whole number has two square roots, one positive and one negative
...
A whole number whose square root is also a whole number is called a perfect square
...
Clearly, there are lots of whole numbers
that are not perfect squares
...
Using our geometric intuition,
it is easy to believe that there are geometric squares whose areas are not perfect square numbers
...
√
2
Area=4
3
5
Area = 5
Area = 9
Page 28
√
The side √
length of the middle square is a number √
which, when squared, yields 5, that is, 5
...
In general, if a whole number lies between two perfect squares, its square root must lie between the
two corresponding square roots
...
Since 21 lies between the perfect squares 16 and 25, 21 must lie between 16 = 4 and
√
25 = 5
...
Between what two consecutive whole numbers does 53 lie?
Solution
...
1
...
4
√
53 must lie between
Exercises
Find the square roots:
√
1
...
81
√
3
...
121
√
5
...
19
√
7
...
26
√
9
...
Complete the square-root table:
√
=6
√0 =
=7
√1 =
=8
√4 =
9=
=9
=4
= 10
=5
= 11
1
...
Moreover, it is
Page 29
easy to see that there is something “left over,” namely, 8
...
The number we start with (100 in the example)
is called the dividend, and the number we repeatedly subtract (23 in the example) is called the divisor
...
1
...
1
Quotient and Remainder
Unlike the other three operations (addition, subtraction, multiplication), the result of a division of whole
numbers consists of not one but two whole numbers: the number of subtractions performed (4 in the
example), and the number left over (8 in the example)
...
Whole number divisions with remainder 0 are called exact
...
Exact divisions can be restated in terms of multiplication
...
On the other hand, starting at 0 and adding 6 (8
times) returns 48
...
In general,
a÷b =c
and
a=b·c
are equivalent statements
...
Express the statement 72 = 8 · 9 as an exact division in two ways
...
We can get to 0 by starting at 72 and repeatedly subtracting 8 (9 times), or by repeatedly
subtracting 8 (9 times)
...
Page 30
Example 24
...
Express this fact using an appropriate multiplication
...
Since the quotient is 9 and the remainder is 0, we can write
63 ÷ 7 = 9
1
...
2
or
63 = 9 · 7
...
Here is what it looks like:
quotient
divisor
dividend
***
***
***
***
***
remainder
The horizontal lines indicate subtractions of intermediate numbers; there is one subtraction for each
digit in the quotient
...
In long division, we try to estimate the number of repeated subtractions that will be needed, multiply
this estimate by the divisor, and hope for a number that is close to, but not greater than, the dividend
...
If it is too small, the
result of the subtraction will be too big – there was actually “room” for further subtraction
...
23 × 3 = 69, which also
leaves a remainder that is too big
...
Now 23 × 4 = 92
...
To check our calculations, we verify that
23 × 4 + 8 = 100
...
)
Let’s try another simple division problem
...
Find the quotient and remainder of the division 162 ÷ 42
...
Putting the dividend and divisor into the long division form, we have
42
162
Let’s estimate the number of subtractions that we’ll need to perform
...
But 4 × 42 = 168, which is too big (bigger than the dividend)
...
We get 3 × 42 = 126, which is less than the dividend, so this must be the
right choice
...
Thus, the quotient is 3 and the remainder is 36
...
(Multiplication before addition
...
The next example shows how to
break the problem down by considering related, but smaller dividends
...
Find the quotient and remainder of the division 3060 ÷ 15
...
If we read the dividend from the left, one digit at a time, we get, successively, the numbers 3,
30, 306, and 3060
...
, but we do not need to think this way
...
To indicate
this we put, for the first digit of the quotient, the digit 2, directly over the right-most digit (0) in 30:
2
15
3060
Now we compute the product 15 × 2 = 30 and subtract it from the initially selected part of the dividend,
i
...
, from 30:
2
15
3060
−3 0
0
In this case, we get a remainder of 0
...
We treat this intermediate remainder
as if it were a new dividend
...
To get one, we
“bring down” the digit 6:
2
15
3060
− 30
06
Now 06 = 6 (why?), and 15 does not go into 6
...
We indicate
this by putting a 0 directly above the digit 6 in the dividend:
Page 33
20
15
3060
− 30
06
Then we bring down the next (and the last) digit (0) of the dividend:
20
15
3060
− 30
060
This gives us a new dividend of 060 = 60, and 15 goes into 60 four times
...
204
15
3060
− 30
060
Multiplying 15 by 4, we subtract this product from 60
...
Page 34
204
15
3060
− 30
060
−
60
0
We have reached the end of our dividend – there are no more digits to bring down
...
The quotient is 204, and the remainder is 0
...
1
...
3
Exercises
Express each exact division as an equivalent multiplication
...
88 ÷ 22 = 4
2
...
51 ÷ 17 = 3
4
...
96 ÷ 12 = 8
Find the quotient and remainder of each division
...
37 ÷ 11
7
...
3007 ÷ 110
9
...
3456 ÷ 241
11
...
578 ÷ 19
13
...
907 ÷ 201
15
...
6
Order of operations
We often do calculations that involve more than one operation
...
Which do we do first? If we do the multiplication first, the
result is 1 + 6 = 7, and if we do the addition first, the result is 3 × 3 = 9
...
It could have been otherwise, but the convention in this case is:
multiplication before addition
...
The precedence of
multiplication can be made explicit using the grouping symbols () (parentheses):
1 + (2 × 3) = 1 + 6 = 7
...
Thus grouping symbols can be used to force any desired order of operations
...
The square root symbol
symbol
...
√
The
symbol acts like a pair of parentheses, telling us to evaluate what is inside (in this case, the
sum 4 + 5) first, before taking the square root
...
operations within grouping symbols first;
2
...
multiplications and divisions (in order of appearance) next;
4
...
“In order of appearance” means in order from left to right
...
Page 36
Example 27
...
Solution
...
There are no grouping symbols, exponents or roots
...
The computations are as follows:
6·5−4÷2+2=
30 − 4 ÷ 2 + 2 =
30 − 2 + 2 =
28 + 2 =
30
In the next two examples, we use the same numbers and the same operations, but we insert grouping
symbols to change the order of operations
...
Example 28
...
Solution
...
After that, the usual order of
operations is followed
...
Evaluate 6 · 5 − 4 ÷ [2 + 2]
...
Now the grouping symbols (brackets) force us to do the addition first
...
The strategy here is to
work from the inside outward
...
Evaluate 62 − [3 + (3 − 1)]2
...
We have parentheses within brackets
...
(That is what “from the inside outward” means
...
Next, the bracketed group, [3 + 2] is evaluated, yielding
62 − 52
...
All that is left is the remaining subtraction,
36 − 25 = 11
...
Evaluate
(11 − 5)2 + (24 ÷ 2 − 4)2
...
We evaluate the two inner groups (11 − 5) and (24 ÷ 2 − 4) first
...
Thus the expression is reduced to
62 + 82
...
√
is a grouping symbol, we evaluate the group 36 + 64 = 100, and finally,
Then, remembering that
evaluate the square root
√
100 = 10
...
6
...
1
...
16 · 4 − 48
3
...
2 · 6 + 2( 36 − 1)
5
...
(2 · 5)2
√
7
...
[18 ÷ (9 ÷ 3)]2
9
...
7
Average
The average of 2 numbers is their sum, divided by 2
...
In general, the average of n numbers is their sum, divided by n
...
Find the average of each of the following multi-sets of numbers
...
) (a) {10, 12}; (b) {5, 6, 13}; (c) {8, 12, 9, 7, 14}
...
In each case, we take the sum of all the numbers in the multi-set, and then (following the
order of operations), divide by number of numbers:
(a) (10 + 12) ÷ 2 = 11;
(b) (5 + 6 + 13) ÷ 3 = 8;
(c)
(8 + 12 + 9 + 7 + 14) ÷ 5 = 10
...
If all the numbers in the multi-set were the same, the average would be that number
...
This gives us another way to define the average of a multi-set of n numbers: it is that number which,
added to itself n times, gives the same total sum as the sum of all the original numbers in the multi-set
...
7
...
1
...
{13, 13, 19, 15}
3
...
{85, 81, 92}
5
...
The number of cancelled
games in the 2002-2009 seasons were 5, 6, 2, 10, 9, 4, 6, 5
...
Suppose the average of the multi-set {20, 22, N, 28} is 25, where N stands for an unknown number
...
Page 39
1
...
“Closed” means that the line segments
form a boundary, with no gaps, which encloses a unique “inside” region, and separates it from the
“outside” region
...
Which of the following figures are polygons?
(a)
(b)
(c)
(d)
(e)
Solution
...
(c) is not a polygon because
it does not have a unique ”inside” region
...
There are two useful numerical quantities associated with a polygon: the perimeter, which is the
length of its boundary, and the area, which is (roughly speaking) the “amount of space” it encloses
...
Areas are measured using square units, such as square feet (ft2 ), square inches
(in2 ), square meters (m2 ), square centimeters (cm2 )
...
Example 34
...
Assume the lengths are measured in feet
...
Adding the lengths of the sides, we find that the perimeter of polygon (a) is
3 + 4 + 1 + 5 + 5 = 18 ft
...
Page 40
Finding the area of a polygon can be complicated, but it is quite simple if the polygon can be divided
up into rectangles
...
The square corners – also known as right angles – imply that the paired opposite
sides have the same length
...
l
w
width = w
length = l
The small squares in the corners are there to indicate that the polygon is a rectangle
...
Estimate the area of the rectangle below, if the square in the upper left corner is 1 unit
on a side
...
If the corner square is 1 unit on a side, the area of the rectangle is the number of squares of
that size that fit inside the rectangle
...
The width (w ) is evidently 3 units, and the length (l ) 7 units
...
7
3
width = 3
length = 7
The example illustrates a general fact: the area of a rectangle is the product of the length and
Page 41
the width
...
It is also clear that the perimeter of a rectangle is the sum of twice the length and twice the
width, since to travel all the way around the boundary of the rectangle, both the length and the width
must be traversed twice
...
Example 36
...
Find the area and
perimeter of the polygon
...
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
Solution
...
For example, the
left-hand vertical side is 7 cm long (stretching from 1 to 8 on the scale at left), and the top side is 2
cm long
...
From this, it follows easily that the perimeter P is given by the sum
7 + 2 + 5 + 4 + 2 + 6 = 26 cm
...
(We could have drawn a different line to obtain a different division into two rectangles
– do you see it?)
Page 42
7
R1
R2
2
2
4
It is evident that R1 has length 7 cm and width 2 cm
...
Similarly, since the length of R2 is 4 cm and the width is 2 cm, the area of R2 is
A2 = 2 · 4 = 8 cm2
...
Thus
A = 14 cm2 + 8 cm2 = 22 cm2
...
The grid is not necessary, as long as we are given enough side-lengths, and assume square corners
...
Find the area and perimeter of the right-angled (square-cornered) polygon below
...
Page 43
5
2
6
3
1
5
Solution
...
We can visualize the polygon as a large
rectangle of length 6 ft
...
5
2
6
3
“bite”
1
5
The area of the large rectangle (bite included) is 6 · 5 = 30 ft2 , and the area of the bite alone is 3 · 2 = 6
ft2
...
To find the perimeter, we need the missing side lengths
...
The left edge is 6 ft, and so the total length of all the right vertical edges must also
be 6 ft
...
So its length must be
6 − (1 + 3) = 2 ft
...
Adding up all the side lengths, starting
(arbitrarily) with the bottom edge and proceeding counter-clockwise, yields the perimeter:
5 + 1 + 2 + 3 + 2 + 2 + 5 + 6 = 26 ft
...
Right triangles are interesting in their own right, and we often consider them in isolation, without
reference to the rectangle they came from
...
The two shorter sides are called the legs
...
c
a
b
A famous formula, called the Pythagorean theorem, states that, in any right triangle with legs of
length a and b, and hypotenuse of length c, the following relation holds:
a2 + b 2 = c 2
...
c = a2 + b 2
...
If A
denotes the area, and the lengths of the legs are a and b as in the figure, the formula is
A = (a · b) ÷ 2
...
Find the area and perimeter of a triangle whose legs have length 3 feet and 4 feet
...
Putting a = 4 and b = 3, we find the area using the formula
A = (a · b) ÷ 2 = (4 · 3) ÷ 2 = 12 ÷ 2 = 6 ft2
...
Thus the perimeter is a + b + c = 4 ft +3 ft +5 ft = 12 ft
...
Example 39
...
The large squares of the grid measure
1 centimeter (cm) on a side
...
Drawing two vertical lines, we can divide up the polygon into two right triangles, T1 and T2 ,
and a rectangle, R
...
Both triangles have legs
of length 1 and 2, so the area of each is (2 · 1) ÷ 2 = 1 cm2
...
To find the perimeter, we use the Pythagorean theorem to find
the lengths of the two slanted sides:
√
c = a2 + b 2 = 12 + 22 = 5
...
√
an
Since 5 is not √ integer, we do not simplify the expression further
...
1
...
1
Exercises
1
...
2
...
3
...
4
...
5
...
Between what two whole
numbers does the answer lie?
6
...
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
Page 47
Page 48
Chapter 2
Fractions and Mixed Numbers
Fractions are expressions such as
3
19
or more generally
a
,
b
where a and b are whole numbers, and b = 0
...
The term on top
is called the numerator, and the term on the bottom is called the denominator
...
Sometimes to save space, we write fractions in one line, using
a slash instead of the fraction bar, putting the numerator on the left and the denominator on the right:
a
= a/b
...
Example 40
...
If the denominator is 2 or 3, we say “halves” or “thirds” (not twoths or threeths!), respectively
...
”
Example 41
...
” 4/3 is spoken “four thirds
...
”
Sometimes, instead of using the th suffix, we just say “a over b
...
”
2
...
In the fraction a/b, the denominator b represents
the number of equal parts into which the whole has been divided, and the numerator represents how
many of those parts are being taken into account
...
49
×
×
×
×
2
3
We can divide any convenient figure into equal parts (not just a rectangle), to represent a fraction
...
So
the picture represents the fraction 3
...
Use a square to represent the fraction 5/16
...
Since 16 is a perfect square, it’s easy to make a square 4 units on a side, and divide that into
16 small squares of equal size
...
2
...
A proper fraction represents
less than one whole
...
Page 50
Example 43
...
The other two are improper
...
The last (10/9) represents a number that is greater than 1
...
For example,
1=
9
159
2
= =
...
For any whole number a (except 0),
a
= 1
...
For example, to represent the whole number
3, we can think of three whole rectangles, each “divided” into 1 “part,” and write
3
3=
...
1
2
...
1
Zero as Numerator and Denominator
The whole number 0 can be written as a fraction in infinitely many ways:
0=
0
,
b
for any non-zero b
...
Thus,
0
0
0
0 = = = =
...
Can it be the denominator? The answer is no
...
For these reasons we attach no meaning to a fraction with denominator 0, saying,
if it comes up, that it is undefined
...
There is another reason why 0 cannot be the denominator of a fraction
...
Suppose we could assign a numerical value to the fraction 1/0, say, 0 = 1
...
But of course, 0 · 1 = 0, so we arrive at 1 = 0, an obvious contradiction
...
2
...
What improper fraction does the following picture represent?
+
+
Use rectangles, circles, or squares to represent the following fractions
...
3
2
3
...
5
8
5
...
4
3
7
...
Write five fractions which are equal to 1
...
Write five fractions which are equal to 0
...
Write 15 as a fraction
...
3
Mixed Numbers
We have seen that an improper fraction represents either a whole number or a whole number plus a
proper fraction
...
Even though it is a sum, the + sign
is omitted
...
3
...
2
Converting an improper fraction into a mixed or whole number
Suppose we have an improper fraction
numerator
...
Then we use these to build the mixed number as follows: the
quotient is the whole number part; the remainder over the divisor is the fractional part
...
Summarizing:
numerator
remainder
= quotient
denominator
divisor
Example 44
...
Solution
...
The corresponding mixed number
has whole number part 4 and fractional part 1/3
...
3
3
If the remainder is 0, the mixed number is actually just a whole number
...
Write the improper fraction 84/7 as a mixed number
...
Dividing 84 by 7 yields a quotient of 12 and a remainder of 0
...
Here are some everyday uses of mixed numbers
...
Five hikers want to share seven chocolate bars fairly
...
For fairness, each of the five hikers should get the same amount: exactly one fifth of the
chocolate
...
5
5
Each hiker gets 1 2 chocolate bars
...
Julissa waited 5 minutes for the train on Monday, 2 minutes on Tuesday, and then 4, 8,
and 3 minutes on Wednesday, Thursday, and Friday, respectively
...
Recall that the average of a set of numbers is their sum, divided by the number of numbers
...
5
5
5
Her average wait time was 4 2 minutes
...
3
...
2
...
4
...
Use
19
3
11
2
135
5
99
98
77
5
mixed numbers to answer the following questions:
6
...
How many sandwiches does each boy get?
7
...
During the first seven days of January, Bill travels 31, 37, 46, 31, 77, 50 and 40 miles respectively
...
3
...
The key fact, again, is
that the fraction
a
a
is always equal to 1, for any number a except 0
...
3
Example 48
...
Solution
...
3 3
It follows that the mixed number 2 1 can be written
3
3 3 1
+ +
...
+
1=
3
3
+
1=
3
3
=
7
3
1
3
(The figure also demonstrates that fractions with the same denominator add up to a new fraction, with
the same denominator, and a numerator which is the sum of all the old numerators
...
)
The procedure in the previous example is easily turned into a general formula
...
q
q
Remember to follow the order of operations (multiplication before addition) when evaluating N · q + p
...
Convert the mixed number 8 3 into an improper fraction, using the boxed rule
...
8
8·3+2
24 + 2
26
2
=
=
=
...
Write the whole number 8 as an improper fraction in three different ways
...
We can think of the whole number 8 as the “mixed” number 8
boxed rule,
8=
0
for any nonzero q
...
q
q
Picking three nonzero numbers for q, say, 2, 5 and 10, we get
8=
2
...
4
16
8·2
= ,
2
2
8=
8·5
40
= ,
5
5
and
8=
8 · 10
80
=
...
1
1
2
2
...
15
4
...
11
5
6
Using N =
N ·q
for any nonzero q, write three fractions equal to each given whole number:
q
6
...
11
8
...
4
Multiplication of Fractions
Fractions are numbers in their own right, and we will recall how to add, subtract, multiply and divide
them
...
We understand what it means to take, let’s say, a third of something: divide it into three pieces,
and take one of the pieces
...
What if the “something” is itself a fraction? For example, what is three fourths of
2
? It is some fraction, but which one?
3
We’ll try to make sense of this question by representing fractions as shaded portions of rectangles
...
3
Divide the same rectangle into 4 equal horizontal parts, and shade 3 of them (using a different shading)
...
By counting, you can see that it
4
3
consists of 6 out of 12 smaller rectangles
...
4
3
12
Example 51
...
Start with a rectangle divided into 3 equal vertical parts, and shade one of the parts, to
represent the fraction 1/3
...
The rectangle is now cut into 2 · 3 = 6 equal parts,
and the two shadings overlap in precisely 1 of the 6 parts
...
2
3
6
c
a
The two examples illustrate a simple rule for finding a “fraction of a fraction,” or of
...
Each of the
Page 57
a shaded vertical rectangles is divided into d parts of which c are differently shaded
...
The rule is
a
b
c
d
of
=
a·c
...
3 5
Example 52
...
8 7
Solution
...
8 7
8·7
56
It is easy to see that fraction multiplication is just an expanded version of whole number multiplication, since every whole number is a fraction with denominator 1
...
Example 53
...
Solution
...
3 1
3·1
3
Example 54
...
How many women are on the jury?
Solution
...
There are four women on the jury
...
So it is correct (and simpler) to
say that
3 2
1
· =
...
We’ll explain
this, and examine some other technical details regarding the representation of fractions, in the next
sections
...
Page 58
2
...
1
Exercises
Find the following products
...
)
1
...
3
...
5
...
7
...
5
1 5
·
3 7
One half of one half
Two thirds of one third
3 3
·
4 4
5
3·
8
2
3
2
1 7
· ·3
2 8
Equivalent Fractions
Fractions which look very different can represent the same number
...
What property do all these fractions share? Each has a denominator that
2
is exactly twice its numerator; the simplest fraction with this property is 1
...
Thus, for example,
1
50
=
...
We give it here because it is so simple
and pleasing, but we postpone the explanation until we discuss proportions
...
In words: two fractions are equivalent if (and only if) their cross-products are equal
...
Starting with a given fraction, we can generate equivalent fractions easily, using the fact that 1 is
the multiplicative identity, and that
c
1=
c
for any non-zero c
...
b
b·c
2
Example 55
...
Solution
...
Picking two values for c, say, 6 and 7, we get two fractions equivalent to 2/3:
2
2·6
12
2
2·7
14
=
=
and
=
=
...
2
...
1
Cancellation and Lowest Terms
The boxed rule above produces equivalent fractions with higher (larger) terms
...
If both numerator and denominator have a
common factor – a whole number greater than 1 which divides them both with zero remainder – we
can “cancel” it by division
...
For example, the numerator and denominator of 24 have the common factor 6
...
24
24 ÷ 6
4
a
a÷c
=
...
It is often indicated as
follows:
3
b
&
3
18
18
&
=
4 = 4
...
To ensure accuracy, you can show the common factor explicitly, before cancelling it
...
24
4·6
4
!
¡
4·6
¡
Page 60
28
70
...
Find lower terms for the fraction
Solution
...
Still lower terms are possible, since 4 and 10 have the common factor 2:
b
&
4
&
2
b
&
10
&
=
5
2
5
(cancelling 2)
...
For example, 2 and 5 have no common factor
other than 1, so the fraction 2 , and also the improper 5 , are in lowest terms
...
Reduce the fraction
24
36
to lowest terms
...
A common factor of the numerator and denominator is 4, so we can reduce the terms as
follows:
6
b
&
24
24
6
&
=
=
(cancelling 4);
9
36
9
b
&
36
&
6 and 9 have the common factor 3, so we can further reduce
2
!
¡
6
2
6
¡
= 3 =
9
3
!
¡
9
¡
(cancelling 3)
...
3
Note that this reduction to lowest terms could have been accomplished in one step, had we realized,
at the outset, that both 24 and 36 are divisible by 12:
2
b
&
24
2
24
&
=
=
3
36
3
b
&
36
&
Example 58
...
to lowest terms
...
A common factor of the numerator and denominator is 13, so that
13
1 · 13
1
=
=
...
Reduce the fraction
Solution
...
= 5
...
5
...
2
5
5
...
for any nonzero c, write four fractions equivalent to each given fraction:
1
5
4
...
=
1
4
2
...
Convert improper fractions to mixed numbers
...
12
8
8
...
20
45
10
...
54
108
12
...
6
Prime Factorization and the GCF
Reducing a fraction to lowest terms requires recognizing a common factor (greater than 1) of the
numerator and denominator
...
In this section, we develop a systematic way of finding the greatest common factor
(GCF) of a set of numbers
...
The first few
prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
...
(This fact is not obvious but was proved
more than two thousand years ago by Euclid
...
(Note that 1 is a special case according to these definitions: it is neither prime
nor composite!) The first few composite numbers are
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30,
...
For example, 15 has factors 3 and 5, that is, 15 = 3 · 5
...
For example:
4=2·2
= 22
8=2·2·2
= 23
6=2·3
9=3·3
10 = 2 · 5
12 = 2 · 2 · 3
14 = 2 · 7
15 = 3 · 5
16 = 2 · 2 · 2 · 2
= 32
= 22 · 3
= 24
(2
...
Start by testing the
number for divisibility by the smallest prime, 2 (a number is divisible by 2 if its final digit is even:
0, 2, 4, 6 or 8)
...
We then repeat the procedure, starting with the last quotient
obtained, and using the next larger prime, 3 (a number is divisible by 3 if the sum of its digits is divisible
by 3 – did you know that?)
...
At that point, we are almost finished
...
The product of all these numbers is
the prime factorization
...
Find the prime factorization of 300
...
300, being even, is divisible by 2, so we start by dividing by 2
...
The primes that were used as divisors were 2, 2, 3, 5
...
The final prime quotient is 5
...
We can easily check our work by multiplying the prime factors and verifying that the product obtained
is the original number
...
6
...
Check the results by multiplication
...
60
2
...
81
4
...
85
6
...
Which of the numbers above is divisible by 3?
2
...
2
Finding the GCF
Using prime factorizations, it is easy to find the greatest common factor of a set of numbers
...
Find the greatest common factor (GCF) of the two-number set {a, b}, where a and b
have the following prime factorizations
...
Solution
...
Note that 7 and 11
are not common, and therefore cannot be factors of the GCF
...
A small power of any prime is a factor of any larger power of the same prime
...
It follows that the GCF is the product of the two smallest powers of 2
and 3
...
Notice that the actual values of a and b (which we could have determined by multiplication) were not
used – only their prime factorizations
...
find the prime factorization of each number;
2
...
if there are no common prime factors, the GCF is 1; otherwise
4
...
the GCF is the product of the common prime factors with the exponents
found in step 4
...
(a) Find the GCF of the set {60, 135, 150}
...
Solution
...
the prime factorizations are
60 = 2 · 2 · 3 · 5 = 22 · 3 · 5
135 = 3 · 3 · 3 · 5 = 33 · 5
150 = 2 · 3 · 5 · 5 = 2 · 3 · 52
2
...
(does not apply to this example);
4
...
the GCF is 31 · 51 = 15
...
The smallest
exponent on all three factors is 1
...
Can you explain why the GCF in part (b) is bigger than in part (a)?
2
...
3
Exercises
Find the GCF of each of the following sets of numbers:
1
...
{72, 48, 36}
3
...
{48, 36}
5
...
{36, 14}
7
...
6
...
Thus,
B
60
60
2
¨¨
=
5 = 5
150
B
¨
¨
150
¨
2
Recall that this is short-hand for
(cancelling the GCF, 30)
...
150 ÷ 30
5
Page 65
60
150
to lowest terms in one
Example 63
...
Solution
...
The smallest exponent on 2 is 2, and the smallest exponents on 3 and 7 are 1; it follows that the GCF
is 22 · 3 · 7 = 84
...
6
...
Exercises
Reduce each fraction to lowest terms by finding and cancelling the GCF of the numerator and denominator
...
1
...
48
36
3
...
36
14
5
...
72
84
7
...
96
56
9
...
300
360
Page 66
2
...
But there’s a short-cut: Since multiplication is commutative,
we can reverse the order of multiplication in the numerator, obtaining
15 21
15 · 21
=
·
...
The trick of cancelling
2
before multiplying – pre-cancellation – saves us from bigger numbers,
315
21 15
·
=
,
5 14
70
and the extra work of finding the GCF{315, 70} = 35 for cancellation:
B
¨9
315
9
¨¨
=
2
2
B
70
¨¨
(cancelling 35),
The general rule is this:
In a product of fractions, a factor which is common to one
of the numerators and one of the denominators can be cancelled before multiplying
...
Notice that pre-cancellation works with any number of factors
...
Find the product
3 8 10
· ·
...
The numerator of the first fraction (3) has a common factor with the denominator of the
third fraction (9), so the product is equal to
1
b
&
3
& 8
·
4
10
·
5 &
b
&
9
3
=
1 8 10
· ·
...
5 3
Finally, the numerator of the third fraction (10) has a common factor with the denominator of the
second fraction (5), so (omitting the factor 1) the product is equal to
2
b
&
5
&
1
·
b
&
10
&
3
2
=
2 2
·
...
The final answer is now a simple product
2 2
4
1
· = =1 ,
1 3
3
3
which is already in lowest terms
...
Mixed numbers are multiplied by simply converting them into improper fractions
...
Find the product 2 8 · 1 1 · 2 2
...
4
3
Solution
...
8 4 3
Cancelling 8’s, we have
85
1
19 · 5
=
=7
...
A gas tank with a 13 1 -gallon capacity is only one third full
...
We need to find the product
1
3
1
· 13 1
...
3
2
3 2
Cancelling the common factor 3,
1
1
Converting
9
2
·
b
&
27
&
9
=
9
...
2
Page 68
27
2 ,
we have
2
...
1
Exercises
Find the products, using pre-cancellation where possible
...
Express any improper fractions as mixed numbers
...
4 7
·
5 12
2
...
12 ·
5 2
·
8 3
4
...
2
of 24
...
2
3
of of 50
...
2 · 1
...
3
8
...
A 12 2 -gallon fish tank is only three-fifths full
...
2 2 cups of beans are needed to make 4 bowls of chili
...
8
Adding and Subtracting Fractions
If I eat a third of a pizza for lunch, and another third for dinner, then I have eaten two thirds in total
...
3 3
3
Similar logic applies whenever we add two (or more) fractions with the same denominator – we simply
add the numerators, while keeping the denominator fixed:
a b
a+b
+ =
...
c
c
c
Fractions with the same denominator are called like fractions
...
Here are three examples involving addition or subtraction of like fractions:
1 2
1+2
3
+ =
=
5 5
5
5
13 − 2
11
4
13 2
− =
=
=1
7
7
7
7
7
7 5
7−5
2
1
− =
= =
...
2
...
1
Exercises
Add or subtract the like fractions as indicated
...
Change improper fractions to mixed numbers
...
1 3
+ =
5 5
2
...
8
11 13
+
+
=
15 15 15
4
...
4
5
−
=
13 13
6
...
109 11
−
=
7
7
8
...
10 6 11
+ −
=
7
7
7
10
...
8
...
4), it looks a bit strange
...
c
a d
c b
a
+ = · + ·
b d
b d
d b
ad
bc
=
+
bd
bd
ad + bc
=
bd
Example 68
...
5 7
Solution
...
5 7
5·7
35
35
Example 69
...
16 24
Solution
...
16 24
16 · 24
384
Cancelling the GCF, 8, we obtain the sum in lowest terms,
83
...
Use the boxed rule to show that
Solution
...
5 5
5
20 + 15
35
7
4 3
+ =
=
=
...
(A needless complication!)
Page 71
A disadvantage of the boxed is that it produces a fraction which may be far from lowest terms
...
To remedy
these, we develop a method, somewhat analogous to pre-cancellation in multiplication, which produces
the sum of any set of fractions in a form which is as close as possible to lowest terms
...
Example 71
...
3 5
Solution
...
5
5·3
15
2 1
10
3
+ =
+
...
15 15
15
15
Since equivalent fractions represent the same number, we conclude that
2 1
13
+ =
...
The multiples of 3 are
3, 6, 9, 12, 15, 18, 21, etc
...
,
and it is easy to see that the smallest number that is a multiple of both – the least common multiple –
is 15
...
8
...
Example 72
...
Solution
...
,
the multiples of 10 are
10, 20, 30, 40,
...
It is evident that the smallest number which is a multiple of all three numbers is 30
...
This is useful when the numbers are rather
large
...
Find the LCM of the two-number set {a, b}, where a and b have the following prime
factorizations
...
Solution
...
The prime factors of a and b
are 2, 3, 7 and 11
...
The LCM is the product of these powers:
LCM{a, b} = 24 · 34 · 7 · 11
...
Notice that the
actual values of a and b, and their LCM (which we could calculate by multiplication) were not needed
– only their prime factorizations
...
find the prime factorization of each number;
2
...
the LCM is the product of the prime factors with the exponents found in
step 2
...
There are similarities
and significant differences
...
8
...
Use the prime factorization method for larger numbers
...
LCM{25, 10}
2
...
LCM{10, 15, 25}
4
...
LCM{9, 6, 12}
6
...
LCM{51, 34, 17}
8
...
LCM{18, 8}
10
...
LCM{4, 14}
12
...
LCM{32 · 52 · 11,
2
...
5
34 · 72 }
The LCD
To add unlike fractions so that the sum is in lowest terms, we use the LCM of their denominators
...
Example 74
...
8
10
18
Solution
...
Looking at the prime factorizations
8 = 23 ,
18 = 2 · 32 ,
10 = 2 · 5,
and taking the highest power of each prime that occurs, we see that the LCM is
23 · 32 · 5 = 360
...
Example 75
...
8 10 18
Solution
...
Now we observe that
360
Thus
It follows that
=
1
1 · 45
=
,
8
8 · 45
8 · 45
=
10 · 36
3
3 · 36
=
10
10 · 36
and
=
18 · 20
...
18
18 · 20
1
3
1
1 · 45
3 · 36
1 · 20
+
+
=
+
+
8 10 18
8 · 45 10 · 36 18 · 20
45 + 108 + 20
=
360
173
=
...
Page 74
(2
...
2
14
− , using the LCD
...
Example 76
...
The LCD is LCM{25, 10} = 50
...
So
14
14 · 2
28
=
= ,
25
25 · 2
50
and
Thus, the difference of the two fractions is
2
2·5
10
=
=
...
50 50
50
50
The latter fraction is not in lowest terms, since the GCF of 18 and 50 is 2
...
25
50
25
b
&
50
&
Example 77
...
3 4
Solution
...
Changing both fractions to equivalent fractions with denominator 12, we get
1·4
4
1
=
=
3
3·4
12
Thus,
and
1
1·3
3
=
=
...
3 4
12 12
12
12
The latter fraction is already in lowest terms, so we are done
...
Find the sum
4 3
+ , reduce to lowest terms, and express the answer as a mixed number
...
The LCD{4, 5} = 20, so that
4 3
4·4 3·5
16 15
31
11
+ =
+
=
+
=
=1
...
8
...
1
...
7 13
+
5
3
3
...
1 0
+
5 2
5
...
11 13
+
15 25
7
...
5
4
−
3 13
9
...
11 11
−
7
17
11
...
6 11
−
7 17
13
...
9
Comparison of Fractions
When two fractions have the same denominator, it is clear that the larger fraction is the one with the
larger numerator
...
Then the largest fraction is the one with the largest numerator,
the second-largest is the one with the second-largest numerator, etc
...
Which is larger, or ?
3
4
Solution
...
4
12
and
The larger of the two fractions with denominator 12 is the one with the larger numerator,
3
that 4 is larger than 2
...
It follows
The inequality symbols < (less than) and > (greater than) are used to state size comparisons
between numbers
...
Visualizing the number line yields a good rule
Page 76
“Left is Less”
if
M
M is left of N
on the number line:
M
N
<
|
|
|
|
|
|
|
|
|
|
|
−5
−4
−3
−2
−1
0
1
2
3
4
5
It is a happy coincidence that the symbol < for “Less than” looks rather like the letter “L
...
(This is the left-to-right ordering on the number line
...
7
3 2
Example 80
...
4 3
9
Solution
...
9
36
We are asked to arrange the fractions in increasing order, so we list them in increasing order of their
numerators, separated by the inequality symbol <
...
36
36
36
Now, going back to the equivalent fractions in lowest terms, we have
2
3
7
< <
...
Arrange the numbers 1 , 1 , 1 , and
in decreasing order
...
Recall that a proper fraction is less than 1, so 11 is obviously the smallest of the four numbers
12
(the others being mixed numbers greater than 1)
...
To compare 3 , 3 ,
5 4
6
and 7 , we find the LCM{5, 4, 7} = 140, and write equivalent fractions with denominator 140:
84
3
=
5
140
3
105
=
4
140
6
120
=
...
The corresponding ordering of the fractions is
6
3
3
> > ,
7
4
5
Page 77
and it follows that the corresponding order of the mixed numbers (each with whole number part 1) is
3
3
6
1 >1 >1
...
7
4
5
12
2
...
1
Exercises
1
...
9 8
3
2
...
5 4
7
3
...
Is 9
5
inches closer to 9 or 10 inches?
8
5
...
10
5
3
to 5
...
How many times does
into” 3 , for example? Equivalently, what fraction results from the division problem
4
2
3
“go
3 2
÷ ?
4 3
It is not at all obvious
...
2
...
1
Reciprocals
Two non-zero numbers are reciprocal if their product is 1
...
The rule for multiplying fractions, together with obvious cancellations, shows that
1
b
&
a b
a
1·&
b
· b
· =
=
1
1
b a
b
&
b ·
a
b
&·1
This means that
Page 78
1
=
1
= 1
...
b
a
(both a and b nonzero)
The reciprocal of the fraction
Since every whole number n can be written as the fraction n , we have the following special case:
1
The reciprocal of the whole number n (nonzero)
1
is the fraction
...
The reciprocal of 5 is 5
...
1
9
is 9
...
For example, the reciprocal of
2/3 is 3/2, and in turn, the reciprocal of 3/2 is 2/3
...
)
• the reciprocal of a number less than 1 is a number greater than 1, and vice versa (since the
reciprocal of a proper fraction is an improper fraction)
...
2
...
2
Exercises
Find the reciprocals of the following numbers
...
1
...
11
3
...
1
25
5
...
10
...
4 3
8
9
÷
12 12
Page 79
We want to know how many times 8 things of a certain size (1/12) go into 9 things of the same size
...
In this form
9
it seems obvious that the answer is the quotient of the numerators, 8
...
We have
shown that
3 2
3·3
3 3
÷ =
= ·
...
We have derived a simple rule:
a
c
a d
÷ = ·
b d
b c
In words: the quotient of two fractions is the product of the dividend and the reciprocal of the divisor
...
Perform the division
5
12
5
÷ 6
...
We convert the division into multiplication by the reciprocal of 6
...
2
Example 84
...
14
We can divide mixed numbers, too, by simply converting them into improper fractions first
...
Perform the division 2 8 ÷ 1 2
...
9
Solution
...
2 ÷1 =
9
2
9
2
Page 80
3
Now we convert the division into multiplication by the reciprocal of 2 :
=
26 2
·
...
3
Example 86
...
The question calls for division, since we need to know how many times 1 3 “goes into” 35
...
4
1
4
1 7
Cancelling 7, we get
20
5 4
· =
= 20
...
2
...
4
Exercises
Perform the divisions
...
1
...
15 ÷
3
...
10 ÷ 5
2
6
5
...
100 50
÷
34
17
7
...
100
÷5
3
Use division to answer the following questions
...
How many 1 1 inch pieces can be cut from a board that is 36 inches long?
2
1
10
...
A gardener needs 32 ounces of seed
...
11
Mixed Numbers and Mixed Units
Mixed numbers can be added and subtracted by first converting them into improper fractions, and very
often this is what we do
...
If we convert to improper fractions, the numerators of both fractions
would be quite large and unwieldy
...
11
...
We must ensure that the resulting mixed
number is in proper form (the fractional part must be a proper fraction), and this involves a procedure
analogous to “carrying” in whole number addition
...
Add 2 1 + 6 3
...
We first align the whole number and fractional places vertically:
2
+
6
1
2
3
4
To add the fractional parts, we write them both with their LCD, which is 4
...
2
+
6
8
2
4
3
4
5
4
The resulting mixed number, 8 5 , is not in proper form, because 5 = 1 1
...
4
(We indicate the carrying by writing {1} above the whole number column
...
Example 88
...
5
Solution
...
Here the convenient choice is 5
...
5
Aligning whole number parts vertically, we have
7
−
3
0
5
2
5
Now we see that the subtraction in the fractions place is not possible (we can’t take 2 fifths from 0
fifths), so we need to borrow 1 from the whole numbers column
...
Borrowing 1 = 5 from 7 (reducing it to 6),
5
5
0
and adding 1 = 5 to 5 at the top of the fractions column, we have
5
6
−
3
3
5
5
2
5
3
5
Both of the previous examples could have been done by first converting the mixed numbers into
improper fractions
...
7−3 =
5
5
5
5
5
Perhaps you find this easier
...
2
Example 89
...
9
Page 83
Solution
...
If we had done the previous example by first converting the mixed numbers into improper fractions, we
would have had to work with rather large numerators:
6891 3035
−
45
45
Not impossible, but the other method is easier!
2
...
2
Exercises
Add or subtract the mixed numbers as indicated
...
1 1
1
...
3 + 7 + 1
5
3
5
1
3
...
75 + 91
4
3
Page 84
5
...
3 − 2
8
4
2
4
7
...
3 + 1 +
5
2 10
1 3
9
...
21 − 19
5
4
6 11
11
...
5
11
4
−4
20
5
2
...
3
Measurements in Mixed Units
When a number results from a measurement, it is given in terms of a unit of measure, or unit for short
...
All 1-foot rulers are divided into 12 smaller units called inches (in), so that
1 ft = 12 in
...
For example,
the length of your table might be 3 feet 6 inches or 3 ft 6 in
...
The
abbreviations are hr, min and sec
...
Mixed units are just like mixed numbers, because the smaller units are simple fractions of the larger
ones:
1
1
1
1 in =
ft
1 min =
hr
1 sec =
min
...
The
notion of proper form remains since, for example, a measurement of
4 ft 17 in
makes sense, but is properly written as
5 ft 5 in,
since any measurement of over 12 inches is at least 1 foot
...
Page 85
Example 90
...
Hugo’s movie is 2
hours and 5 minutes long, while Hector’s is 1 hour and 43 minutes
...
Hector’s waiting time is the difference in the lengths of the movies,
2 hr
5 min
− 1 hr
43 min
The subtraction in the minutes column can’t be done, so we borrow 1 hr = 60 min from the hours
column, reducing the number of hours at the top of the hours column from 2 to 1, and increasing the
number of minutes at the top of the minutes column from 5 min to 5 min + 60 min = 65 min
...
Here’s an example involving length measurements
...
Two boards are laid end-to-end
...
What is the total length of the boards? Give your answer in proper form
...
We need to calculate the mixed-unit sum
6 ft
8 in
+ 11 ft 10 in
Adding the inches column, we obtain 18 in = 1 ft 6 in
...
Page 86
6 in
2
...
4
Exercises
1
...
What is the length of the leftover piece?
2
...
A film critic watched three movies
...
(a) What was the total length of the movies? (b) What was the average length of the
movies?
2
...
operations within grouping symbols first;
2
...
multiplications and divisions (in order of appearance) next;
4
...
We remind you that “in order of appearance” means in order from left to right, and that “grouping symbols” include parentheses, brackets, braces (curly brackets), the square root symbol, and the horizontal
a+b
fraction line as in c−d
...
Calculate 2 − +
...
Subtraction comes first in the left-to-right order
...
10
Page 87
10
5 ,
(LCD = 10)
we have
Example 93
...
5 2
Solution
...
Rewriting 3 and 2 with the LCD 10, we have
5
5
6
+
10 10
2−
Example 94
...
10
=2−
2 1
+1 ·
...
We evaluate the expression with an exponent first (( 2 )2 = 4 ), followed by the mixed number
3
9
multiplication, followed by the addition
...
18
(converting the mixed number to an improper fraction)
(cancelling 5)
(LCD = 18)
1 2
1
1
Example 95
...
4 3
6
2
Solution
...
We first convert mixed numbers into improper fractions, and change the division into multiplication by the reciprocal
...
18
Page 88
(lowest terms)
(LCD = 18)
1
1
1 2
Example 96
...
Solution
...
In this case, we must work “from the inside out
...
From the previous example we know that
1
1
1
1 2
1 · −1 ÷1 = ,
4 3
6
2
18
thus we get
1
1
1 2
1+ 2+ 1 · −1 ÷1
4 3
6
2
= 1+ 2+
2
...
1
1
18
= {1 +
55
1
37
}=
=3
...
Express improper fractions as mixed numbers
...
3 2 4
+ ÷
4 3 9
2
...
3
1
+3
4
2
−2
4
...
7 ÷ + 1 · 2
2 5
8 5
6
...
Find the perimeter of a rectangle with length 2 1 ft and width 1 3 ft
...
Julia has two closets
...
The floor of the other is a
2
rectangle of width 3 1 ft and length 4 1 ft
...
2
2
How much will it cost if the tiles are $5 per square foot?
Page 89
Page 90
Chapter 3
Decimals and Percents
Decimals are mixed numbers in which the proper fractional part has a denominator which is 10, or 100,
or 1000, or, more generally, a power of 10
...
The number above and to the right of 10 is the exponent, or power, and indicates the number of times
that 10 appears as a factor
...
Here are some examples of mixed numbers in which the denominator of the fractional part is a power
of ten, together with their representations as decimals:
3
10
37
21
100
21
13
1000
1
=
1
...
37
=
13
...
The digits to the right of the decimal point represent the fractional
part according to the following rule:
• the digits to the right of the decimal point constitute the numerator;
• the number of digits to the right of the decimal point is the power of 10 which constitutes the
denominator
...
As we shall see, this makes computation very easy
...
In the example
21
13
= 13
...
3
...
Thus, you will recall that
4267 stands for
4 thousands + 2 hundreds + 6 tens + 7 ones
...
The decimal point allows us to adjoin more places (to the right of the decimal point), with place values
less than 1
...
59 stands for
2 tens + 3 ones + 5 tenths + 9 hundredths
...
More generally,
The nth place to the right of the decimal point has the place value
1
...
If we adopt the convention that
1
10−n = n ,
10
then the sequence of place values, from left to right, is just the sequence of decreasing powers of ten:
...
, thousands, hundreds, tens, ones
(decimal point) 10−1 ,
10−2 ,
10−3 ,
...
Example 97
...
671
...
The tens (101 ) place is the second place to the left of the decimal point
...
The hundredths (10−2 ) place is the second place to the right of the decimal point
...
Example 98
...
00286?
Solution
...
The digit in that
place is 8
...
1
...
19
...
0
...
98
...
Name the digit in the hundred-thousandths place in the decimal 1
...
5
...
6
...
3
(b) 28
...
1
(d) 4
...
2
Significant and Insignificant 0’s
Decimals are read aloud in the following way:
• read the whole number part;
• read “and” for the decimal point;
• read the digits to the right of the decimal point as if they represented a whole number;
• append the place-name of the last decimal place
...
(a) 1
...
”
(b) 31
...
(c) 48
...
”
In the decimal 48
...
006 stands for 0 tenths + 0 hundredths + 6 thousandths
...
” Nonetheless, the two 0’s in
...
6
If we had written
...
006, we would have named a different fraction, namely, 10 , rather than
6
1000
...
23, the first two 0’s are unnecessary, since the whole
number 006 is just 6
...
How do we know which 0’s are significant, and which are insignificant? We can precede any whole
number or whole number part of a decimal by as many 0’s as we want, because these 0’s do not change
the value of the number (as in 006 = 6)
...
For example,
...
98 because
...
In these cases,
the 0’s are insignificant
...
Page 93
304
3004
Example 100
...
3004 is not equal to 2
...
34, because the fractional parts, 10000 , 1000 , and
34
100 are not equal
...
3004 are significant, and
therefore cannot be omitted
...
All other 0’s are significant, and cannot be omitted
...
In the decimal 0304
...
Thus
0304
...
908
...
Does 23400 contain any insignificant 0’s?
Solution
...
The decimal point is not shown, but is “understood” to be at the
rightmost end
...
It is sometimes convenient to preserve or adjoin non-significant 0’s
...
Thus the proper fraction 100 =
...
67
...
3
Comparing Decimals
Insignificant 0’s are also useful in comparing decimals as to size
...
It is
very easy to find a common denominator for two or more decimals, because the number of decimal
places to the right of the decimal point is all that is needed to determine the denominator: if there are
n places to the right of the decimal point, the denominator is 10n
...
There is no computation involved: we simply “pad
out” the shorter decimals with insignificant 0’s to the right of the decimal place until all have the same
number of decimal places
...
Example 103
...
102, 0
...
2
...
Of the three given decimals, 0
...
So we pad out all three to 5 places, using insignificant 0’s: Thus
0
...
10200
0
...
09876
0
...
20000
Page 94
(3
...
102 =
10200
100000
0
...
2 =
20000
...
Hence we have
0
...
102 > 0
...
3
...
1
Exercises
Eliminate the insignificant 0’s in the following decimals
...
210304
...
00206
...
210
...
21030900
Arrange each group of decimals in descending order, from largest to smallest:
5
...
2,
...
121
...
1
...
9, 1
...
7
...
106, 0
...
61
...
9
...
14, 9
...
099
...
4
Rounding-off
The numbers
0
...
11,
0
...
1111,
0
...
After a few steps,
they are almost too close together to visualize
...
Such small quantities can be important in the sciences, but even in circumstances where precision is
important, there is always a limit beyond which small differences become negligible – not worth worrying
about
...
) Being precise, but not overly precise, is
the purpose of rounding-off numbers
...
That means choosing the place
whose value we consider the smallest worth worrying about
...
In the first instance, we would round off our dollar amounts
to the nearest hundred, and in the second, to the nearest hundredth (cent)
...
$175 is closer to $200 than to $100, so, rounding to the nearest hundred, we say that
an annual salary of $36, 175 is approximately $36, 200
...
The symbol ≈ means “approximately equal to
...
16
...
80 than to 16
...
84 ≈ 16
...
Notice that we dropped the insignificant 0 in 16
...
Example 106
...
135 is exactly halfway between 0
...
140, so, if we want to round to the nearest
hundredth, it is not clear whether we should round “up” to 0
...
130
...
The convention is that “halfway is almost home,” so we round up
...
135 ≈ 0
...
140
...
To round off a decimal to a given place (the round-off place):
1
...
Replace all digits to the right of the round-off place by 0’s;
3
...
Example 107
...
03 to the nearest tenth
...
The round-off place is the tenths place, and the right-neighboring digit (3) is less than 5, so
we preserve the digit in the round-off place (0) and replace all digits to the right (there is only one) with
0
...
00
...
Thus
26
...
0 (to the nearest tenth)
Example 108
...
Solution
...
The right neighbor of the round-off digit is 5, so, by the “halfway” convention,
we round the 0 in the round-off place up to 1, and replace all digits to the right with 0
...
Notice that the two 0’s in 100 are significant, and cannot be eliminated!
Page 96
One final remark about step 2 of the procedure
...
Example 109
...
597 to the nearest hundredth
...
The 9 in the round-off place has right-neighbor 7, which is greater than 5
...
597 ≈ 6
...
3
...
1
Exercises
Round off each number twice: (a) to the nearest ten; (b) to the nearest hundredth
...
304
...
96
...
115
...
100
...
7
...
5
Adding and Subtracting Decimals
One of the advantages of decimals over ordinary fractions is ease of computation
...
The
only question is where to put the decimal point
...
This puts the decimal points in alignment as well
...
Carrying and borrowing
are done as with whole numbers
...
Find the sum of 96
...
03
...
Line up the numbers vertically so that the ones places (and hence the neighboring decimal
points) are directly on top of eachother
...
8
+ 3 4 2
...
(This is not strictly necessary: just remember that an empty place is
occupied by a 0
...
9 8
...
0 3
...
8 0
+ 3 4 2
...
8 3
The sum is 440
...
Example 111
...
09, 56
...
205
...
Line up the numbers vertically at the decimal points (equivalently, at the ones place)
...
Add
the columns from right to left, imagining 0’s in the empty places, and carrying when necessary
...
0 9
5 6
...
+
0
...
6 7 9
Subtraction follows the same vertical alignment rule
...
Find the difference 50 − 4
...
Solution
...
We fill it out with insignificant 0’s so that both the
minuend and the subtrahend have the same number of decimal places
...
0 0 0
4
...
Borrowing from the tens place of the subtrahend, we can write
50
...
= 4 tens + 9 ones + 9 tenths + 9 hundredths + {10} thousandths
= 49
...
9 9 {10}
4
...
2 9
Page 98
4
3
...
1
Exercises
Find the sums or differences:
1
...
48 + 56
...
804
...
8409
3
...
09 − 32
...
4
...
38409
5
...
093 − 6
...
100 − 23
...
830 − 16
...
80 − 56
...
Verify the subtraction problems above by doing an appropriate addition
...
Find the sum of 64
...
3, 4, and 9
...
11
...
6 and 0
...
6
Multiplying and Dividing Decimals by Powers of 10
What happens when we multiply a whole number by 10? Ones become tens, tens become hundreds ,
hundreds become thousands , etc
...
(In the
product, there are no longer any ones , and that fact must be recorded with a significant 0 in the ones
place
...
Making the decimal point explicit in our example,
234
...
We can describe what happens this way: when we multiply by 10, all the digits shift one place to the
left, including the insignificant 0 which was understood to be in the tenths place of the whole number
234
...
) Similarly, if we multiply a
whole number by 100, all the digits shift two places to the left, including the two insignificant 0’s
which are understood to be in the tenths and hundredths places
...
00 × 100 = 59 700
...
Thus, for example,
281 × 104 = 2 810 000
...
Here are some examples:
38
...
3
0
...
0031 × 1000 = 3
...
09 × 104 = 120 900
100 × 102 = 10000
...
It is often convenient to imagine that, when multiplying by a power of 10, the digits remain fixed
while the decimal point moves (to the right )
...
) This description leads to a very easy rule for multiplying a decimal by a positive power
of 10:
To multiply a decimal by 10n , move the decimal point n places to the right
...
The whole discussion above can be repeated, except that, in this case
of division, digits shift to the right , or, equivalently, the decimal point moves to the left
...
Here are some examples:
623 ÷ 10 = 62
...
023 ÷ 100 = 0
...
5 ÷ 103 = 0
...
For example,
5 ÷ 100 = 5 × 10−2 = 0
...
5 ÷ 10 = 6
...
65
86
...
37 × 10−4 = 0
...
00008
Notice that the left movement of the decimal point is indicated by the negative exponent
...
6
...
1
...
48 × 10
2
...
48 ÷ 10
3
...
804
...
804
...
0
...
0
...
38 × 10−3
9
...
09 × 104
10
...
09 ÷ 105
11
...
3 × 10−3
12
...
7
Multiplication of general decimals
We already know how to multiply and divide decimals when one of them is a power of 10
...
Consider the
following example
...
Multiply 0
...
07
...
Writing this as a product of ordinary fractions, we get
0
...
07 =
3
7
3×7
21
×
=
=
...
021
...
3 × 0
...
021
...
By looking back at the previous section, it is easy to see the
product of two powers of 10 is itself a power of 10
...
Thus any set of decimals can be multiplied by following
a two step procedure:
To multiply two or more decimals:
• Multiply the decimals as if they were whole numbers, ignoring the decimal
points (this gives the numerator of the decimal fraction);
• Add the number of decimal places in each factor (this gives the denominator
of the decimal fraction by specifying the number of decimal places)
...
Find the product 21
...
004
...
Temporarily ignoring the decimal points, we multiply 2102 × 4 = 8408
...
02 has two
decimal places, and 0
...
So the product will have 2 + 3 = 5 decimal places
...
02 × 0
...
08408
...
Find the product of 12, 0
...
004
...
Ignoring the decimal points, the product is 12 × 3 × 4 = 144
...
Thus,
12 × 0
...
004 = 144 ÷ 104 = 0
...
3
...
1
Exercises
Find the following products
...
68
...
804 × 6
...
26
...
004
4
...
09 × 93
5
...
9
6
...
093 × 6
...
64
...
345
8
...
0001 × 0
...
1000 × 0
...
6 × 0
...
02 × 0
...
8
Division of a decimal by a whole number
If the divisor is a whole number, the division procedure is almost exactly like the long division procedure
for whole numbers
...
For example, to set up the division 12
...
0342
Then we put a decimal point vertically aligned with the decimal point in the divisor, where the quotient
will go:
...
0342
Now the long division is performed with no further attention paid to the decimal point
...
Continuing, we estimate that 31
goes into 120 3 times, so 3 is placed in the next (tenths) place in the quotient, and 3 × 31 is subtracted
from 120 to yield 27
...
)
0
...
0342
−9 3
2 7
We bring down the next digit in the dividend, which is 3, and start over with a new dividend of 273
...
8 × 31 = 248, which is subtracted from 273 to yield 25
...
38
31
12
...
The remaining steps are as follows
...
3882
31
12
...
0342 ÷ 31 = 0
...
We can check the result by multiplication, of course:
31 × 0
...
0342, as it should
...
In fact, because of the decimal point, we never need to write
remainders again
...
Example 116
...
4 ÷ 5
...
We set up the long division process as before, with a decimal point where the quotient will
go, directly above the decimal point in the dividend
...
5
13
...
6
5
13
...
6
5
13
...
6 8
5
13
...
Thus 13
...
68
...
But this doesn’t always happen
...
In this case, we get what
is know as a repeating decimal
...
Perform the division 3
...
Solution
...
Page 105
...
2
11 goes into 32 twice with a remainder of 10:
...
2
− 2 2
1 0
Adjoin an insignificant 0 to the dividend, bring it down, and perform another step: 11 goes into 100 9
times with a remainder of 1
...
29
11
3
...
We record that with a 0 digit in the quotient, adjoin another insignificant 0 to the dividend,
and bring it down
...
Page 106
...
20
− 2 2
1 00
−
99
100
−
99
1
It is evident that these last two steps will now repeat again and again, forever
...
Still, it is easy to describe the quotient: it will consist
of 29 after the decimal point, followed by an endless string of 09’s
...
2909090909090909 · · · = 0
...
Thus, 3
...
2909
...
For example, the repeating decimal
0
...
is written, using the bar notation, as
0
...
The repeated string can be quite long, or just a single digit
...
428571
2 ÷ 11 = 0
...
6
Sometimes, we don’t care whether a decimal terminates or repeats
...
Example 118
...
To the nearest cent, how much did you
spend for each button?
Solution
...
Since we are rounding to the nearest hundredth
(cent), we need only carry out the division as far as the thousandths place
...
000 ÷ 900
...
467, which, rounded to the nearest hundredth, is 0
...
Therefore, the buttons cost approximately 47 cents each
...
46777777 · · · = 0
...
Page 107
3
...
1
Exercises
Perform the indicated divisions
...
If you round off, indicate
to what place
...
91 ÷ 20
2
...
68 ÷ 5
3
...
86 ÷ 71
5
...
02 ÷ 9
6
...
19 ÷ 13
7
...
09 ÷ 215
8
...
17 ÷ 19
10
...
8 ÷ 40
3
...
61 ÷ 2
...
Just multiply both the dividend and divisor by a power of 10 sufficiently large
to make the divisor into a whole number, and then proceed as before
...
5 turns into the whole number 25 and the dividend turns into 86
...
Recall that two fractions are equivalent, that is, represent the same number, if one is obtained from the
other by multiplying both numerator and denominator by the same nonzero number
...
Thus,
8
...
1
8
...
2
...
5 × 10
25
In other words, the division problem 8
...
5 has the same quotient as the division problem 86
...
We do the latter problem exactly as in the previous section
...
444
25
86
...
8
...
5 = 3
...
(Question: why
didn’t we write 3
...
Perform the division 32
...
41 and round off the quotient to the nearest hundredth
...
There are two decimal places in the divisor, 6
...
41 × 102 = 641
...
7 ÷ 641
...
We therefore add two insignificant 0’s to the dividend, and perform the division as
follows:
5
...
700
− 3205
1 700
− 1 282
418
Since the digit in the thousandths place of the quotient is less than 5, we preserve the digit in the
hundredths place
...
00 (to the nearest hundredth)
...
9
...
1
...
4 ÷ 1
...
0
...
4
3
...
09 ÷ 9
...
9
...
0215
Perform the divisions
...
4
...
5
6
...
09 ÷ 0
...
353 ÷ 2
...
0
...
002
9
...
997 ÷ 0
...
10
Percents, Conversions
A percent is a fraction in which the denominator is 100
...
For example,
97%
means
97
100
or
0
...
Percents need not be whole numbers, and they need not represent proper fractions
...
5
100
0
...
5% =
= 0
...
A decimal can be converted to a percent by moving the decimal point two places to the right and
adjoining the % symbol
...
Thus
0
...
05 = 205%
0
...
8%
1
...
0067 = 0
...
Thus
92% = 0
...
2% = 0
...
38
71
...
7102
To convert a percent to a fraction (or mixed number), first convert the percent to a decimal, as
above, then express the decimal as a fraction (with a visible denominator), and finally, reduce the fraction
to lowest terms, if needed
...
Convert 10
...
Solution
...
8% as a decimal
...
8% = 0
...
Then, we write the decimal as a fraction with a visible denominator
...
0
...
1000
Page 110
Finally, the GCF of the numerator and denominator is 4, so we cancel 4 from both of these numbers,
obtaining a fraction in lowest terms:
X
$$
108
27
108$
=
250 = 250
...
1 =
1
,
10
20% = 0
...
25 = ,
4
1
50% = 0
...
2
To convert a fraction to a decimal, we just perform long division
...
Convert
1
4
to a decimal
...
Performing the division 1 ÷ 4, we obtain
0
...
00
− 8
20
−20
0
Thus
1
4
= 0
...
To convert a fraction to a percent, first convert it to a decimal, and then convert the decimal to a
percent
...
Convert
1
40
to a percent
...
Performing the division 1 ÷ 40, we obtain 0
...
Then, we convert 0
...
We obtain
1
= 2
...
40
3
...
1
Exercises
Convert the following percents to decimals
...
43%
2
...
56
...
4
...
5
...
09
6
...
00679
7
...
384
8
...
384
Convert the following decimals or percents to fractions (or mixed numbers) in lowest terms
...
44%
10
...
0
...
0
...
40%
14
...
98%
16
...
2%
17
...
Round percents to the nearest tenth
of a percent
...
1
8
19
...
2
5
21
...
3
4
23
...
1
12
25
...
11
Fractional parts of numbers
We now have several ways of indicating a fractional part of a number
...
25 of 90
25%
of
90
are all different ways of describing the number 1 × 90 = 22
...
The word “of” indicates multiplication
...
The fractional part taken need not be a proper fractional part
...
Example 123
...
Solution
...
25, and then multiplying by 500, we get
1
...
1
Example 124
...
What is the sales tax on a shirt priced at $25?
Solution
...
25, and 8
...
0825
...
0825 × 25 = 2
...
06
...
Example 125
...
If he earns $1250 per month, how much does he
5
have left over, after paying his rent?
Solution
...
So he has
5
X
$ 250
$$
1250
left over after paying rent
...
11
...
Find 16% of 75
2
...
Find
...
Find 150% of 105
5
...
A $500 television is being sold at a 15% discount
...
Angela gets a 5% raise
...
What is her new salary?
8
...
If the car originally sold for $12, 500, what it would
5
it sell for now?
9
...
If there were 24 problems on the test, how
10
...
If Maribel’s medical expenses were $550, and her total income was $28, 000, can she deduct her
medical expenses?
Page 114
Chapter 4
Ratio and Proportion
In this chapter we develop another interpretation of fractions, as comparisons between two quantities
...
1
Ratio
If a team wins ten games and loses five, we say that the ratio of wins to losses is 2 : 1, or “2 to 1
...
5
1
We can do the same for any two quantities, a and b, as long as b = 0
...
Example 126
...
Solution
...
7
21
=
12
4
or
7 : 4
...
(c) The ratio of 64 to 4 is
Notice that in (c) we left the denominator 1 (rather than just writing 16) to maintain the idea of a
comparison between two numbers
...
When the ratio is expressed in lowest terms, however,
it is always the ratio of two whole numbers, as small as possible
...
What is the ratio of 3 4 to 1 1 ?
2
115
Solution
...
× =
3 ÷1 =
2
1
4
2
4
3
2
b 3
& &
b
&
4
&
The ratio is 5 : 2
...
What is the ratio of 22
...
We could perform the decimal division 22
...
150
2
The ratio is 3 : 2
...
Otherwise the ratio will be a skewed or false comparison
...
Find the ratio of 15 dollars to 30 cents
...
If we form the ratio
= , we imply that a person walking around with $15 in his pocket
30
2
has half as much money as a person walking around with only 30 cents! The proper comparison is
obtained by converting both numbers to the same units
...
The
correct ratio is
50
1500 cents
=
...
Manuel gets a five minute break for every hour he works
...
Since 1 hour = 60 minutes, the ratio of work time to break time is
11
55 minutes
=
...
” So a percent can be viewed as a ratio whose second
term (denominator) is 100
...
At a certain community college, 55% of the students are female
...
Solution
...
For (b), we first deduce that
45% of the students are male (since 45% = 100% − 55% )
...
45
9
Page 116
Example 132
...
Text is
2
8
printed between the margins
...
The total margin width is 5 + 5 = 5 = 1 4 inches (taking into account both left and right
8
8
4
margins
...
2
4
4
The ratio of the width of the printed text to total margin width is
1
1
29 5
29 4
29
7 ÷1 =
÷ =
· = ,
4
4
4
4
4 5
5
or 29 : 5
...
1
...
1
...
30 to 32
3
...
1 8 to 3 1
4
1
5
...
14
...
4
7
...
69 to 2
...
3 hours to 40 minutes
1
9
...
In the late afternoon, a 35 foot tree casts an 84 foot shadow
...
2
Proportions
A proportion is a statement that two ratios are equal
...
1
Page 117
A proportion is a statement of the form
c
a
=
b
d
where b, d = 0
...
2
...
Example 133
...
20
5
Solution
...
The general cross-product property is stated below for reference:
a
c
=
b
d
if and only if ad = bc
...
An equation is a
mathematical statement of the form X = Y
...
N
N
a
c
= , and multiply both sides by the nonzero number bd
...
b
d
Cancelling b on the left and d on the right yields the cross-product property, ad = bc
...
Example 134
...
(a)
2
4
6
3
= ; (b)
=
...
(a) The cross-products are 3 · 7 = 21 and 2 · 11 = 22
...
(b) The cross-products are 4 · 15 and 10 · 6, both equal to 60
...
Page 118
4
...
2
Solving a proportion
If one of the four terms of a proportion is missing or unknown, it can be found using the cross-product
property
...
In the proportion below, x represents an
unknown term (any other letter would do)
...
3
51
There is a unique x which makes the proportion true, namely, the one which makes the cross-products,
51x and 34(3), equal
...
51
51
Cancellation yields
1
b
&
51x
&
b
&
51
&
=
1
B
¨2
102
¨¨
b
&
51
&
= 2
...
It doesn’t matter which of the four terms is missing; the proportion can always be solved by a similar
procedure
...
Solve the proportion
36
9
=
100
y
for the unknown term y
...
The cross-products must be equal, so
9y = 3600
...
The unknown term is 400
...
Neither the terms nor the solution of a proportion are necessarily whole numbers
...
Solve the proportion
10
15
=
B
4
for B and check the solution
...
Setting the cross products equal,
40 = 15B
...
3
The unknown term is 8 or 2 2
...
3
?
10(4) =
?
40 =
2
2
(15)
3
b
&
15
&
8
b
&
3
&
1
1
40 = 40
...
Solve the proportion
2
3
5
=
x
...
Set the cross-products equal, and solve for x:
2
· 15 = 5x
3
5
2
b
&
15
1· & = 5x
b
&
3
&
10 = 5x
2 = x
...
Solve the proportion
x
42
=
...
5
Page 120
5
Solution
...
Stringing two proportions together
70
70
5
3
42
x
=
=
5
70
1
...
It is evident that the solution to our original proportion is the
solution to the simpler proportion
x
3
=
...
5
Setting the cross-products equal
4
...
5
we obtain the solution x =
= 0
...
5
We summarize the procedure for solving a proportion
...
Reduce the numerical ratio (not containing the
unknown) to lowest terms, if necessary;
2
...
Divide both sides of the resulting equation by
the number multiplying the unknown term
...
In the original proportion, replace the unknown
term with the solution you obtained;
2
...
Equivalence of fractions, and hence, most of fraction arithmetic, is based on proportion
...
Then
1 1
x +y
+ =
...
Page 121
4
...
3
Exercises
Solve the following proportions
...
1
3
=
5
x
2
...
20
100
=
5
y
4
...
11
1
=
B
2
6
...
s
4
=
3
13
8
...
2
=
7
0
...
75
P
=
100
125
10
...
3
31
4
5
= 1
x
22
Percent problems
Any problem involving percent can be stated (or restated) in the form
“A is P percent of B”
where one of the numbers A, B or P is unknown
...
100
If we divide both sides of the equation by B, we obtain the proportion in the box below
...
B
100
The letter B is used to suggest “base amount” – that is, B is the amount from which (or of which)
a percentage is taken
...
It could be that A (the
percentage taken) is greater than B, but only if the percentage P is greater than 100
...
Example 139
...
The percent is unknown, and the base amount B is 300 (since it follows the word “of”)
...
” The corresponding proportion is
6
P
=
300
100
and the solution (verify it!) is P = 2
...
Example 140
...
The base amount B is 150, since it follows the word “of,” the percentage is P = 8, and therefore A is the unknown amount
...
” The corresponding
proportion is
8
A
=
150
100
and the solution is A = 12 (verify!)
...
(Note that this result could have been obtained without using a proportion, since 8% of 150 simply
means
...
)
Example 141
...
The base amount B is unknown, and A is 290
...
” The corresponding proportion is
58
290
=
B
100
and the solution is
29000
58
= 500
...
Everyday questions involving percent are not always as straightforward as in the previous examples
...
Example 142
...
If there are 2970 female
students, what is the total enrollment at BCC?
Page 123
Solution
...
The question can be restated as “2970 is 55% of B” and the corresponding proportion is
therefore
55
2970
=
...
B=
11
The total enrollment is 5400 students
...
Example 143
...
What was the percent
increase in the tuition? (Round to the nearest tenth of a percent
...
The tuition increased by $3000 − 2850 = $150
...
Equivalently, we want to answer the question “$150 is what percent of $2850?”
We solve the proportion
P
150
=
2850
100
3
P
=
57
100
57P = 300
P ≈ 5
...
3%
...
3
...
12 is 20% of what number?
2
...
12 is 40% of what number?
4
...
90 is what percent of 300?
6
...
What is 125% of 600?
8
...
1)
9
...
)
10
...
A baseball team won 93 games, or 62% of the games it played
...
New York State sales tax is 8
...
If the sales tax on a DVD player is $16
...
Marina’s annual salary last year was $56, 000
...
By what
percent did her salary increase?
14
...
By what percent did the population decrease?
4
...
For example, for any given car,
the ratio of miles driven to gallons of gas used,
miles
gallon
or
“miles per gallon”
is essentially unchanging, or fixed
...
Example 144
...
How many gallons will she need to drive
225 miles?
Solution
...
Before setting the cross-products equal, reduce
30
the fraction on the left side to
...
30
2
(4
...
Miles per gallon is an example of a rate, or comparison of unlike quantities by means of a ratio
...
Other examples of rates are: dollars per hour (pay rate for an hourly worker), dollars per item (price of
an item for sale), calories per minute (energy use by an athlete)
...
Page 125
Example 145
...
5 miles
...
Solution
...
3
...
Then
375
x calories
=
(setting cross-products equal)
3
...
5x
(375)(26)
x=
≈ 2786
...
5
She will burn approximately 2786 calories running the marathon
...
5 miles
calories per mile
...
A painting crew can paint three apartments in a week
...
The ratio
apartments painted
time in weeks
is given as 3
...
and y =
40
3
4
...
1
Exercises
Use proportions to solve the following problems
...
On a map,
the map?
3
4
inch represents 14 miles
...
A truck burns 2 1 quarts of oil on an 1800 mile trip
...
An investment of $2, 000 earns $48 in interest over a year
...
)
Page 126
4
...
Approximately how many bottles would
you expect to be leaking in a sample of 20, 000 bottles?
5
...
If a bar of
aluminum weighs 15 pounds, how much would a bar of lead of the same size weigh?
4
...
Having the
same shape means that the three angles of one triangle are equal to the three corresponding angles in
the other
...
E
95◦
B
95◦
A
50◦
35◦
C
D
35◦
50◦
F
Given any triangle, we can obtain a similar one by enlarging (or reducing) all the side lengths in a fixed
ratio
...
If AB stands for the length of the side AB, etc
...
Equivalently,
DE
EF
DF
2
=
=
=
...
We must be careful to compare side lengths in the proper order
...
We could have used the reverse
order, smaller : larger, but only if we also used the reversed ratio 1 : 2
...
This is the key fact about similar triangles
...
EF
DF
DE
Example 147
...
AB =
15 feet, BC = 12 feet, and DE = 22
...
Find EF
...
C
22
...
F
12 ft
...
E
Solution
...
AB
BC
Filling in the given information, and using x to represent the unknown side length EF , we have
22
...
5)(12) = 15x
270 = 15x
270
x=
= 18
...
We now describe two ways of obtaining a pair of similar triangles
...
Two line segments are parallel if they do not cross, even if extended infinitely in either
direction (think of straight train tracks)
...
If two sides of a triangle are joined by a line segment parallel to the third side, the resulting
(smaller) triangle is similar to the original triangle
...
Both triangles are visible on the right (the smaller overlapping
the larger), so there was really no need to draw the first picture
...
AB
CA
CB
smaller triangle
larger triangle
2
...
C
D
C
D
E
A
B
A
B
Starting with the parallel segments AB and CD, we connected D to A and C to B by means of
straight lines (called transversals) which cross at point E
...
In addition, ∠D = ∠A and ∠C = ∠B
...
) It follows that triangles ABE and CDE are similar and hence
upper triangle
lower triangle
CD
EC
ED
=
=
...
In the picture below, AB
length of the side AE
...
Some of the side lengths are given in inches
...
75
1
...
Because AB
CD,
ABE ∼
CDE
...
AB
EB
EA
smaller triangle
bigger triangle
In particular, using EC = 1, EB = 1
...
75, and x for the length of AE ,
1
...
5
x
or x = (1
...
75) = 1
...
The length of AE is 1
...
Example 149
...
Find the length of BC
...
D
2 ft
...
E
2 ft
...
B
C
Page 130
Solution
...
Corresponding sides are proportional, hence
2 ft
...
4 in
...
2 ft
...
6 in
...
5 in
...
5
...
Find the length
x in each case
...
5
1
...
6
3
x
1
...
1
...
4
C
F
Page 131
D
3
...
Find the side length x
...
5
E
1
...
5
A
x
C
4
...
Find the side length x
...
4 in
...
D
1 ft
...
x
A
2 ft
...
E 1 ft
...
C
5
...
Find the side length x
...
In the figure, DE
BC
...
If BC is 12 centimeters long, how long is DE ?
A
D
E
B
C
7
...
How long is AB if DE has
A
5 meters
D
3 meters
B
E
C
Page 133
8
...
Find the side length x
...
75 in
...
50 in
...
...
E
C
A
9
...
How long is the shadow of a nearby 32 foot
tree? Draw a figure involving similar triangles which illustrates the situation
...
We now include negative
numbers, which lie to the left of 0 on the number line, forming a kind of “mirror image” of the positive
numbers
...
The negative numbers
have sign −, and the positive numbers have sign +
...
0 has no sign
...
Thus, for example,
−3 < −1, since −3 lies to the left of −1 on the number line
...
41
3
3
and −π ≈ −3
...
- 13
3
|
√
- 2
-π
|
|
|
√
|
|
|
13
3
π
2
|
|
|
|
−5 −4 −3 −2 −1
2
3
4
5
0
1
The absolute value of a number is its distance from 0 on the number line
...
Thus | 0 |= 0 and
√
√
√
| −1 |=| 1 |= 1,
| − 2 |=| 2 |= 2,
| −π |=| π |= π, etc
...
This is
because, intuitively, distance is a nonnegative quantity
...
1
Adding signed numbers
To do arithmetic with signed numbers, we extend the ordinary operations of addition, subtraction,
multiplication and division so that they remain consistent with the familiar operations with nonnegative
numbers
...
135
Thus, to add 3 to 2, we imagine starting at 2 on the number line and moving 3 “steps” to the right,
arriving at 5
...
Perform the signed number addition 2 + (−3)
...
We start at 2 as before, but now we move 3 steps left, taking the sign of −3 into account
...
We conclude that 2 + (−3) = −1
...
Example 151
...
Solution
...
Now we start at −3,
and move 2 steps right, taking into account the (positive) sign of 2
...
Addition, extended to signed numbers, remains commutative
...
Perform the signed number addition −1 + (−3) in both possible orders
...
Both numbers are negative, so we move consistently left on the number line
...
Either way,
2
1
we arrive at −4
...
Perform the signed number addition −2
...
75
...
Starting at −2
...
75 to the right, ending at 1
...
+3
...
50
1
...
75 − 2
...
25), while the sign of the sum (+) was the same
as the sign of the number with the larger absolute value
...
Page 137
When two signed numbers are added
• if the numbers have opposite signs,
1
...
the absolute value of the sum is the difference bewteen the two individual absolute
values (larger − smaller)
...
the sign of the sum is the common sign
of the summands;
2
...
Example 154
...
Solution
...
e
...
Thus the sum is −3
...
Example 155
...
Solution
...
The absolute value
of the sum is the sum of the individual absolute values, | −7 | + | −9 |= 7 + 9 = 16
...
The reasoning is summarized as
−7 + (−9) = −(7 + 9) = −16
...
Add −3
...
9
...
The numbers have opposite signs, and the number with the larger absolute value (4
...
The absolute value of the sum is the difference | 4
...
6 |=
4
...
6
...
6 + 4
...
9 − 3
...
3 = 1
...
1
3
Example 157
...
5
3
Page 138
Solution
...
The absolute value of the sum is the difference in the individual absolute
values,
1
3
=7 −4
3
5
9
5
=7 −4
15
15
20
9
=6 −4
15
15
11
=2
...
1
...
15
Exercises
Add
...
−6 + 19
2
...
−34 + (−28)
4
...
5 + −4
5
2
6
...
38 + (−9
...
−1001
...
−
3
4
+2
9
...
3
1
5
+ 2
5
8
Use an appropriate signed number addition for the following
...
Find the temperature at noon in Anchorage if the temperature at dawn was −11◦ F and the
temperature subsequently rose by 36◦ F
...
Find the height (in feet above ground level) of an elevator which started 30 feet below ground level
and subsequently rose 70 feet
...
1
...
Pairs of numbers such as {− , }, {− 2, 2}, {−π, π}, which are unequal but equidistant
3 3
from 0, and hence have the same absolute value, are called opposites
...
) To find the opposite of a nonzero number, we simply
change its sign
...
(a) The opposite of 11 is −11
...
7 is 1
...
Since there are only two possible signs, the opposite of the opposite of a number is the number we
started with
...
Example 159
...
(b) the opposite of the opposite
8
of −5 is −(−(−5)) = −5
...
The rule for adding
signed numbers with opposite signs doesn’t seem to work here: it is not clear what the sign of the sum
should be
...
Accordingly, −3 + 3 = 0
...
) In general,
The sum of a number and its opposite is 0:
N + (−N) = 0
...
(b) −23
...
04 = 0
...
e, the number does not change
...
Example 160
...
−4 + 0 = −4
...
1
...
The opposite of 0
...
The opposite of −2
5
7
Page 140
3
...
The sum of 99 and its opposite
...
The sum of −π and its opposite
...
The opposite of the opposite of −0
...
7
...
8
...
−9
...
1)
3
10
...
0 + 5
...
4 + 2 + −2
3
3
5
...
4
Associativity
Another important property of addition, associativity, extends to signed number addition
...
Recall that this property allowed
us to add long columns of nonnegative numbers
...
Then we add the two subtotals, treating the subtotal associated with the negative numbers
as negative
...
Example 162
...
6 + (−5
...
5 + (−134) + 158
...
3)
Solution
...
6
6 9
...
7
2 7 1
...
8
1 3 5
...
+ 1 6 2
...
1
Page 141
Treating the subtotal associated with the negative numbers as negative, we add 271
...
1)
...
8 + (−437
...
1 − 271
...
3
...
” There is a net gain if the total profits are larger than the total
losses (in absolute value); otherwise there is a net loss
...
Last week, a business received checks from clients in the amounts of $350
...
00
and $900
...
50, $551
...
70
...
Bills paid are losses, so we treat them as negative numbers; checks received are profits, counted
as positive numbers
...
6 5
4 6 1
...
7 8
1, 7 1 2
...
5 0
5 5 1
...
7 0
8 9 1
...
43 + (−891
...
43 − 891
...
23
...
23 for the week
...
1
...
1
...
−8
...
5) + 8
...
44 + (−5
...
8 + 36 + (−19
...
5
...
−1
2
3
+ −2
5
6
+ −8
1
3
7
...
50 + (−21
...
95) + 13
...
99 + (−86
...
5 + −3
1
2
1
1
+4 + −2
2
2
+ (−4) + −
1
2
+7
9
...
10
...
05, $865
...
25, and paid bills in the amounts of $561
...
25, $798
...
25
...
2
Subtracting signed numbers
The phrase “subtract A from B” indicates the operation B − A
...
To subtract a signed number A from a signed number B,
add B to the opposite of A:
B − A = B + (−A)
...
For
example, instead of subtracting 17 − 9, we could simply add 17 + (−9) (verify that the results are the
same)
...
Example 164
...
(b) Subtract 8 from −2
...
(a) 9 − 15 = 9 + (−15)
...
(b) −2 − 8 = −2 + (−8)
...
Example 165
...
Solution
...
Remembering that −(−17) = 17 (opposite of the opposite),
we have
12 − (−17) = 12 + 17 = 29
...
Subtract (−15) from −32
...
−32 − (−15) = −32 + (−(−15)) = −32 + 15
...
Example 167
...
8
6
Solution
...
24
(LCD = 24)
Example 168
...
359 − 10
...
Solution
...
359−10
...
359+(−10
...
08−3
...
We perform the subtraction of absolute
values vertically
...
0 8 0
−
3
...
7 2 1
Remembering that the sign was negative, 3
...
08 = −6
...
Formerly “impossible” subtractions, such as
7 − 12
can now be easily performed
...
Examples like this show that subtraction is not commutative: changing the order of subtraction changes
the result to its opposite
...
Example 169
...
We often need to find the difference of two unequal quantities
...
If the two quantities are A and B, their difference is either A − B or B − A
(whichever is positive)
...
Formally, the difference of A and B is defined to be the absolute value of A − B
...
Find the difference in temperature between −2◦ F and 50◦ F
...
Intuitively,
higher temperature − lower temperature = 50 − (−2) = 52,
so the temperature difference is 52◦ F
...
Example 171
...
Not far away,
in Death Valley, the lowest point is 282 feet below sea level
...
Assigning a negative altitude to a point below sea level, the difference is
14505 − (−282) = 14505 + 282 = 14787 feet
...
2
...
1
...
98 − 100
3
...
4)
4
...
5 from 2
...
Subtract −53 from 68
...
−8
...
11)
7
...
2 from
1
5
8
...
−87 − (−23)
3
10
...
5 − (−11)
8
12
...
1 and express the difference as a fraction
...
−2
...
−
1
2
− −
2
3
15
...
If water boils at 100◦ C , by how much
did the temperature of the snowball rise?
16
...
The elevation of the lake bottom is −66
...
The car is lifted by a crane to a height 79
...
Through what vertical distance
was the car lifted?
5
...
For example,
3 × 4 = 4 + 4 + 4 = 12
...
3 × (−4) = (−4) + (−4) + (−4) = −12
...
The general rule is:
The product of two numbers with opposite signs is the negative of the product of their
absolute values
...
Find the products (a) 7 × (−11) and (b) (−12) × 5
...
We take the negative of the product of the absolute values in each case
...
(b) (−12) × 5 = −(12 × 5) = −60
...
It makes no sense to
“repeatedly” add a number to itself when the number of repeats is negative! How should we define
(−2) × (−3)?
It is best to give up on intuition and let consistency rule
...
Example 173
...
Solution
...
Note that the product of two positive numbers could also be defined as the (positive) product of their
absolute values
...
[OPTIONAL: If you find it hard to accept that the product of two negative numbers is positive,
the following discussion might help
...
That is, for any three numbers, A, B and C ,
A(B + C ) = AB + AC
...
In particular, if A and B are positive,
(−A)(B + (−B)) = (−A)(B) + (−A)(−B)
...
It follows that (−A)(−B) must
be the opposite of (−A)(B) = −(−AB) = AB
...
]
Here is a summary of the rules for multiplying signed numbers
...
• if the numbers have opposite signs, the product is negative
...
In the examples and exercises below, we freely use all three ways of symbolizing multiplication: ·,
×, and juxtaposition
...
Example 174
...
Solution
...
Example 175
...
Page 147
Solution
...
Example 176
...
30)(−2
...
Solution
...
Temporarily ignoring the
decimal points,
630
×
205
3150
1260
129150
Inserting four decimal places counting from the right, and remembering that the product is positive, we
conclude that
(−6
...
05) = 12
...
Example 177
...
4
5
Solution
...
−
3
b
&
4
&
2
b
&
6
&
·
Example 178
...
1
...
The numbers have the same sign so the product is positive, and we compute with the absolute
values
...
2 4
8
8
Recall the zero property, which states that
0 · x = 0 and x · 0 = 0,
for any number x
...
Example 179
...
1) · 0 = 0
...
−
15
31
·1=−
and
x · 1 = x,
15
...
Multiplication, like addition, continues to be associative when extended to signed numbers
...
One
consequence is that the sign of a product of several signed numbers can be quickly determined in advance
by simply determining whether the number of negative factors is even or odd
...
This is true because every pair of negative factors has a positive product
...
But if the number of negative factors is odd, there is always one
“unpaired” negative factor, which makes the total product negative
...
(−1)(2)(−3)(−4) = −24
(−1)(−2)(−3)(−4) = 24
5
...
1
(an odd number (three) of negative factors makes a negative product),
(an even number (four) of negative factors makes a positive product)
...
1
...
−9 × 7
3
...
(−1) × 5
5
...
4)(−1)
6
...
8)
7
...
(−1) × (−1) × (−1)
9
...
5 × (−31)
10
...
03) × (−0
...
7
9
×0
12
...
−1
6
7
−1
1
2
14
...
62)(1000)
15
...
(−3)(−50)(−2)
17
...
5)(−0
...
(3)(−10)(2)(−5)
19
...
5
...
We extend this definition unchanged to
signed numbers
...
Similarly, − and − are reciprocal, since
2
8
3
−
3
8
−
8
3
= 1
...
e
...
Since a number and its reciprocal must have the same sign (why?), it follows that
the rules for dividing signed numbers are exactly analogous to the rules for multiplying them
...
In both cases, the absolute value of the quotient is the quotient of
the individual absolute values
...
By the rule for division of signed
y
numbers, if either x or y (but not both) is negative, the fraction represents a negative number
...
Then
Recall that a fraction bar indicates division, that is,
a
a
−a
=
=−
...
In the examples and exercises, we indicate division by the ÷ symbol or
the fraction bar interchangeably
...
For divisions of whole numbers, explicit conversion to reciprocal multiplication is not
necessary
...
Divide 24 by −6
...
The quotient 24÷ (−6) is negative since the numbers have opposite signs
...
Thus
24 ÷ (−6) = −4
...
Express the fraction
−33
as a mixed number
...
The fraction represents division of numbers with opposite signs, so the result is negative
...
5
5
Example 184
...
6
Solution
...
We convert the mixed
number to an improper fraction and multiply by the reciprocal of the divisor
...
3
Example 185
...
738 ÷ (−100)
...
The quotient is negative because the numbers have opposite signs
...
738 ÷ (−100) = −0
...
(Recall that dividing a decimal by a power of 10 (102 in this case) simply results in a leftward shift of
the decimal point by a number of places equal to the exponent on 10
...
Divide 16 ÷ (−0
...
Solution
...
We can either perform the long
division
0
...
25 to the fraction
1
and multiply by its reciprocal
...
25) = − 16 ÷
5
...
1
1
4
= −(16 × 4) = −64
...
This remains
true for signed fractions
...
A fraction like
makes sense and is equal to 0, so
−3
0 ÷ (−3) =
0
0
= − = −0 = 0
...
(You may wish to review the
0
discussion in Section 2
...
1, where all attempts to make sense of a fraction with denominator 0 led to
failure or contradiction
...
For a nonzero signed number N
•
0÷N =0
•
N ÷ 0 is undefined
...
Example 187
...
62 ÷ 0 and
0
are both equal to 0
...
0
Page 152
5
...
2
Exercises
Perform the divisions, or state that they are undefined
...
(−24) ÷ (−8)
2
...
66 ÷ 0
4
...
(−30) ÷ 6
6
...
−19
0
8
...
9
...
9
10
...
03 ÷ (−1000)
11
...
4
25
÷ −
5
32
13
...
−10 ÷ 5
4
15
...
−110
...
5 ÷
...
50 ÷ (−
...
5
Powers of Signed Numbers
Since exponents indicate repeated multiplication, there is no problem applying exponents to signed
numbers
...
Page 153
Example 188
...
Solution
...
8
8
=−
Example 189
...
2)4
...
(−0
...
2)(−0
...
2)(−0
...
0016
...
)
The previous examples show that
A power of a negative number is
• negative if the exponent is odd,
• positive if the exponent is even
...
For example, the square of −3 is written
(−3)2 = (−3)(−3) = 9
...
Rather, −32 represents the
opposite of 32 , so
−32 = −9
...
Evaluate
−
3
4
2
and −
3
4
2
Solution
...
16
The behavior of 0 in an exponential expression, in particular, the interpretation of 0 as an exponent,
extends unchanged to signed numbers
...
4 on powers of whole numbers
in Chapter 1
...
)
Example 191
...
5)0 = 1
...
5
...
04 = 0
...
1
...
(−8)2
3
...
−63
5
...
05
7
...
−(−33 )
9
...
01)3
11
...
(−9
...
(−23)0
14
...
4)3
15
...
00
Page 155
00 is undefined
...
2
18
...
6
2
4
Square Roots and Signed Numbers
If there is a number whose square is the number n, it is called a square root of n
...
It is also true that (−4)2 = 16, so −4 is another square root of 16
...
√
• The negative square root is denoted − N
...
It has only one square root, itself:
√
0 = 0
...
Example 192
...
The square root of a whole number which is not a perfect square has a decimal expansion which
neither terminates nor repeats
...
) A calculator will inform you that
√
√
2 ≈ 1
...
73
(rounded to the nearest hundredth)
...
Between what two whole numbers do − 2 and − 3 lie? Which number is larger?
√
√
Solution
...
41 and − 3 ≈ −1
...
√
- 3
|
√
- 2
√
|
|
0
−2
−1
Recalling that “left is less” on the number line,
√
√
− 3 < − 2,
√
in other words, − 2 is larger
...
Between what two integers does − 19 lie?
Page 156
2
√
3
|
|
1
2
Solution
...
We find that 19 is between 42 and 52
...
It follows (by a geometric
argument we gave in Section 1
...
3) that
4<
√
19 < 5
...
√
√
− 19
19
|
|
|
|
|
|
|
|
|
|
|
−5
−4
−3
−2
−1
0
1
2
3
4
5
√
√
We see that − 19 lies between −5 and −4
...
The square root of a positive fraction is not usually a quotient of whole numbers
...
For example,
2
2 2 4
4
2
= because
cannot be evaluated in this way
...
But
9
3
3
9
5
√
2
2
=√
5
5
(since
√
2
√
5
2
2
= )
...
5
We come now to a troubling fact
...
Suppose, for example, there were a number which, when squared, yields −4
...
But
22 = (−2)2 = 4
(not −4)
...
It is possible to expand the set of signed numbers
so as to remedy this defect
...
We leave that for a more advanced course
...
√
Example 195
...
Page 157
5
...
1
Exercises
Find the square roots, or state that they are not real numbers
...
9
√
2
...
−25
√
4
...
6
...
0
7
...
−
10
...
−
25
36
−64
−49
1
16
Between what two integers do the following square roots lie?
√
11
...
− 7
√
13
...
√
√
− 8
14
...
12
15
5
...
For example, the expression
x +y
x −y
Page 158
indicates a fraction whose numerator is the sum of two unspecified numbers, x and y , and whose
denominator is their difference
...
Expressions can be evaluated
(assigned a numerical value) if numerical values are assigned to all the variables appearing in the
expression
...
Example 196
...
We replace each letter by its assigned numerical value, enclosed in parentheses,
x +y
(2) + (−6)
=
,
x −y
(2) − (−6)
and simplify the resulting expression
−4
1
(2) + (−6)
=
=−
...
x +y
Example 197
...
x −y
Solution
...
x −y
(−3) − (3)
−6
The reason for replacing letters with their assigned values in parentheses is to avoid pitfalls such as
the ones highlighted in the next two examples
...
Evaluate x 2 if x = −4
...
Forgetting parentheses, we would write x 2 = −42 = −16, which is wrong, since the square of
any nonzero number is positive
...
Example 199
...
Solution
...
The correct evaluation is
ab = (3)(−4) = −12
...
Page 159
1
...
exponents and roots next;
3
...
additions and subtractions (in order of appearance) last
...
1
Example 200
...
2
Solution
...
2
Example 201
...
Solution
...
For the second expression,
a − (b − c) = (2) − (−11) − (10)
= 2 − (−21)
(subtraction within grouping symbols first)
= 23
...
Evaluate 3 − (p − r ÷ t) if p = , r = − and t =
...
Within grouping symbols, division comes first
...
=
=3−
20
20 20
20
20
Example 203
...
x2 + 2
3(0)3 − 5(0)2 + 13(0) − 4
−4
=
= −2
...
Evaluate the expressions a2 + b 2 and a + b if a = −0
...
8
...
Solution
...
6)2 + (0
...
36 +
...
For the second expression,
a + b = −0
...
8
= 0
...
This example shows that, in general,
a2 + b 2 = a + b
...
7
...
Reduce fractions to lowest terms and
express improper fractions as mixed numbers
...
2a − b
2
...
−6x 2
if x = −
4
...
a − ab − b
if a = −0
...
8
6
...
(d − e)(d 2 + ed + e 2 )
8
...
2pq − q 2
10
...
3a − 2b
5a + b
5
...
2 and q = 2
...
−b +
if d = −1 and e = −4
if x =
b 2 − 4ac
1
1
and b =
2
4
1
2
and y = −
3
2
if a = −2, b = 7 and c = −3
Using Formulae
A formula expresses one quantity in terms of others
...
If we know the length and width of a rectangle, we can calculate its area, using the formula
...
Example 205
...
Use the formula to determine the distance an object falls in (a) 2 seconds; (b) 4 seconds
...
We substitute t = 2 and t = 4 into the formula
...
Example 206
...
5
◦ F if a thermometer reads −10◦ C
...
Substituting (−10) for C in the formula, we find
9
F = (−10) + 32
5
=−
9
b
&
5
&
1·
b
&
10
&
1
2
+ 32
= −18 + 32 = 14
...
Example 207
...
c=
a2 + b 2
...
Solution
...
√
√
c = a2 + b 2 = (5)2 + (12)2 = 25 + 144 = 169 = 13 feet
...
The semi-perimeter of a triangle with side lengths a, b and c is given by the formula
1
s = (a + b + c)
...
Solution
...
Thus
a = 80 in,
b = 112 in,
and
c = 142 in
...
Example 209
...
Find the dosage
for a four-year old child if the adult dosage is 48 mg (milligrams)
...
Substituting t = 4 and A = 48,
C=
1 48
4
· 48 = ·
= 12
...
Page 163
5
...
1
Exercises
1
...
8 meters and whose width is 3
...
Use the
formula A = lw , where A is the area, l is the length, and w is the width
...
Find the perimeter of the rectangle in the preceding exercise, using the formula
where P is the perimeter and l , w are the length and width, respectively
...
Find the length of the hypotenuse of a right triangle whose legs are 0
...
4 yards
...
4
...
s = 16t 2 ,
where s is the distance
9
5
...
Use the formula F = C + 32
...
Heron’s formula, A = s(s − a)(s − b)(s − c), gives the area (A) of a triangle (not necessarily
a right triangle) with side lengths a, b and c, and semi-perimeter s
...
2
7
...
Find A after 2 years if the interest
rate is 5% (r =
...
8
...
2 grams
...
C=
t + 12
5
...
If the statement has just one
variable, and if
• the variable does not appear in the denominator of a fraction,
√
• the variable does not appear under a
symbol,
• the variable is not raised to a power other than 1,
then we have a linear equation in one variable
...
3
A solution to an equation in one variable is a number which, when substituted for the variable, makes
a true statement
...
Show that −4 is a solution to the equation −3y = 12
Solution
...
Page 164
Example 211
...
Solution
...
2
Example 212
...
3
Solution
...
When we substitute 4 for t, we obtain
2 4
· =0
3 1
8
2 − = 0,
3
2
− = 0,
3
2−
a false statement
...
A linear equation in one variable x is an equation that can
be written in the form
ax + b = c
for some numbers a, b and c, with a = 0
...
e
...
5
...
1
Finding solutions
We find solutions to equations using two common-sense principles:
• Adding equals to equals produces equals,
• Multiplying equals by equals produces equals
...
Adding equals to equals,
2=2
3=3
2+3 =2+3
produces equals: 5 = 5
...
Although we cannot say that x − 3 and 6 are “equals” (without knowing the value of x) we can say
that if they are equal for some x, then adding equals to both sides, or multiplying both sides by equals,
will produce a new pair of equals for the same x
...
The solution to the last equation is obvious (and obviously unique): x must be 9
...
Could there be some other solution to the original
equation, say, x = p? If so, then p − 3 = 6, and adding 3 to both sides yields p = 9
...
We conclude that 9 is the unique solution of x − 3 = 6
...
Find the solution of the equation 9 = − z by multiplying both sides by equals
...
Solution
...
To check that −45 is the solution, substitute −45 for z in the original equation, and verify that a true
statement results
...
Equations which have the same solution are called equivalent
...
Page 166
To find the solution (solve) a linear equation in one variable,
we can do one or both of the following:
• add the same number to both sides of the equation;
• multiply both sides of the equation by the same nonzero number
...
(This is because
subtracting a number is the same as adding the opposite number, and dividing by a nonzero number is
the same as multiplying by the reciprocal number
...
Solve the equation 2x + 4 = −10 and check that the solution is correct
...
First subtract 4 from both sides:
2x + 4 = −10
−4
−4
2x = −14
...
The solution is −7
...
2x + 4 = −10
?
2(−7) + 4 = −10
?
−14 + 4 = −10
−10 = −10
a true statement
7
Example 215
...
8
7
Solution
...
But it is
8
better not to do that immediately
...
7
26 = 5 − x
8
− 5 − 5
7
21 = − x
8
Page 167
Now we multiply both sides by
−
8
7
8
8
− · 21 = −
7
7
3
8
8
¡
b
&
−
· & = −
21
&
7
7
&
¡
−24 = x
7
− x
8
7
¡
− x
8
¡
A linear equation given in the standard form
ax + b = c
(a = 0)
is solved in two steps: first, subtract b from both sides; second, divide both sides by a
...
Solve the equation −x + 5 = 12
...
Note that −x means −1x
...
−x + 5 = 12
−x = 7
x = −7
(subtracting 5 from both sides)
(dividing or multiplying both sides by −1)
Note that dividing by −1 is the same as multiplying by −1, since −1 is its own reciprocal!
Example 217
...
2
2
Solution
...
2
1
1
=
2
2
1 1
3x = −
2 2
3x = 0
3x +
x = 0
...
Recall that
0
= 0
...
Four times a number is −5
...
What is the number?
Solution
...
The sentence “Four times a number is −5
...
6
...
6
= −1
...
4
The number is −1
...
It is straightforward to check that four times −1
...
6
...
9
...
1
...
6 = 14
...
2x + 3 = 19,
y =2
x =8
Solve the equations and check the solutions
...
x − 5 = 19
4
...
9x = 45
6
...
−y − 14 = 5
8
...
19t − 5 = −5
10
...
2y − 1
...
2
12
...
Twelve times a number is 108
...
1
Five more than a number is −9
Title: Arithmétic exercices !!! for only 50 $ !!!!!
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Description: you need arithmétic exercices ? defintions ? here you are !!!!!