The Rational Numbers

CHAPTER 1. THE ARITHMETIC OF NUMBERS
18
1.3
The Rational Numbers
1. The greatest common divisor of the numerator and denominator is gcd(20, 50) =
10. Divide numerator and denominator by 10.
20
20 ÷ 10
=
Divide numerator and denominator
50
20 ÷ 10
by the greatest common divisor.
=
2
5
Simplify.
3. The greatest common divisor of the numerator and denominator is gcd(10, 48) =
2. Divide numerator and denominator by 2.
10
10 ÷ 2
=
Divide numerator and denominator
48
10 ÷ 2
by the greatest common divisor.
=
5
24
Simplify.
5. The greatest common divisor of the numerator and denominator is gcd(24, 45) =
3. Divide numerator and denominator by 3.
24
24 ÷ 3
=
Divide numerator and denominator
45
24 ÷ 3
by the greatest common divisor.
=
8
15
Simplify.
7. Prime factor both numerator and denominator, then cancel common factors.
153
3 · 3 · 17
=
Prime factorization.
170
2 · 5 · 17
3·3·
17
Cancel common factors.
=
2·5·
17
3·3
=
2·5
9
=
Simplify numerator and denominator.
10
9. Prime factor both numerator and denominator, then cancel common factors.
188
2 · 2 · 47
=
Prime factorization.
141
3 · 47
2·2·
47
=
Cancel common factors.
3·
4
7
2·2
=
3
4
=
Simplify numerator and denominator.
3
Second Edition: 2012-2013
1.3. THE RATIONAL NUMBERS
19
11. Prime factor both numerator and denominator, then cancel common factors.
159
3 · 53
=
106
2 · 53
3·
5
3
=
2·
5
3
3
=
2
Prime factorization.
Cancel common factors.
13. First, multiply numerators and denominators. Then prime factor the
resulting numerator and denominator and cancel like factors. Simplify your
final answer.
18
20
360
· −
Unlike signs yields negative answer.
=−
8
13
104
2·2·2·3·3·5
=−
Prime factor numerator and denominator.
2 · 2 · 2 · 13
2
·2
·2
·3·3·5
=−
Cancel common factors.
2·
2·
2 · 13
3·3·5
=−
13
45
Simplify numerator and denominator.
=−
13
15. First, multiply numerators and denominators. Then prime factor the
resulting numerator and denominator and cancel like factors. Simplify your
final answer.
18
19
342
Like signs yields positive answer.
− · −
=
4
13
52
2 · 3 · 3 · 19
=
Prime factor numerator and denominator.
2 · 2 · 13
2 · 3 · 3 · 19
= Cancel common factors.
2 · 2 · 13
3 · 3 · 19
=
2 · 13
171
=
Simplify numerator and denominator.
26
17. First, multiply numerators and denominators. Then prime factor the
resulting numerator and denominator and cancel like factors. Simplify your
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
20
final answer.
−
16 19
304
·
=−
8 6
48
2 · 2 · 2 · 2 · 19
=−
2·2·2·2·3
2·
2 · 2 · 2 · 19
=−
2
·
2
· 2 · 2 · 3
19
=−
3
Unlike signs yields negative answer.
Prime factor numerator and denominator.
Cancel common factors.
19. Factor the individual numerators and denominators first, cancel common
factors, then multiply numerators and denominators. Like signs gives a positive
answer.
5
12
5
− · −
=−
6
49
2·3
5
=−
2 · 3
5·2
=
7·7
10
=
49
2·2·3
· −
7·7
2 · 2 · 3
· −
7·7
21. Factor the individual numerators and denominators first, cancel common
factors, then multiply numerators and denominators. Unlike signs gives a negative answer.
−
3·7 2·2·3
21 12
·
=−
·
10 55
2 · 5 5 · 11
3 · 7 2 · 2 · 3
·
=−
2 · 5 5 · 11
3·7·2·3
=−
5 · 5 · 11
126
=−
275
23. Factor the individual numerators and denominators first, cancel common
factors, then multiply numerators and denominators. Unlike signs gives a negSecond Edition: 2012-2013
1.3. THE RATIONAL NUMBERS
ative answer.
21
55
54
2·3·3·3
5 · 11
· −
· −
=
29
11
29
11
2·3·3·3
5·
11
· −
=
1
1
29
5·2·3·3·3
=−
29
270
=−
29
25. First, invert and multiply.
5
58
50
50
÷ −
· −
=
39
58
39
5
Factor the individual numerators and denominators first, cancel common factors, then multiply numerators and denominators. Unlike signs gives a negative
answer.
2 · 29
2·5·5
· −
=
3 · 13
5
2 · 29
2 · 5 · 5
· −
=
3 · 13
5
2 · 5 · 2 · 29
=−
3 · 13
580
=−
39
27. First, invert and multiply.
−
60 31
60 34
÷
=− ·
17 31
17 34
Next, factor the individual numerators and denominators first, cancel common factors, then multiply numerators and denominators. Unlike signs gives
a negative answer.
2·2·3·5
·
17
2 · 2 · 3 · 5
·
=−
17
2 · 3 · 5 · 31
=−
17 · 17
930
=−
289
=−
31
2 · 17
31
2 · 17
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
22
29. First, invert and multiply.
7
13
28
7
− ÷ −
=− · −
10
28
10
13
Factor the individual numerators and denominators first, cancel common factors, then multiply numerators and denominators. Like signs gives a positive
answer.
2·2·7
7
· −
=−
2·5
13
2 · 2 · 7
7
· −
=−
13
2 · 5
7·2·7
=
5 · 13
98
=
65
31. Since the denominators are different, start by writing equivalent fractions
using the least common denominator. Then add the numerators over the common denominator and simplify.
5 2 1 3
5 1
− + =− · + ·
6 4
6 2 4 3
=−
3
10
+
12 12
=
−10 + 3
12
=
−7
12
Make equivalent fractions
with LCD = 12.
Simplify numerators and
denominators.
Add numerators over common
denominator.
Simplify numerator.
Although this answer is perfectly acceptable, negative divided by positive gives
a negative answer, so we could also write
=−
7
12
33. Since the denominators are different, start by writing equivalent fractions
using the least common denominator. Then add the numerators over the comSecond Edition: 2012-2013
1.3. THE RATIONAL NUMBERS
23
mon denominator and simplify.
1
1 3
8
8 1
− + −
=− · + − ·
9
3
9 1
3 3
3
8
=− + −
9
9
=
−8 + (−3)
9
=
−11
9
Make equivalent fractions
with LCD = 9.
Simplify numerators and
denominators.
Add numerators over
common denominator.
Simplify numerator.
Although this answer is perfectly acceptable, negative divided by positive gives
a negative answer, so we could also write
=−
11
9
35. First simplify by rewriting as an addition problem. Then, since the denominators are different, write equivalent fractions using the least common
denominator. Finally, add the numerators over the common denominator and
simplify.
2
1
1
− − −
=−
4
9
4
1
=−
4
=−
+
·
2
9
9 2 4
+ ·
9 9 4
8
9
+
36 36
=
−9 + 8
36
=
−1
36
Rewrite as an addition problem.
Equivalent fractions
with LCD = 36.
Simplify numerators and
denominators.
Add numerators over
common denominator.
Simplify numerator.
Although this answer is perfectly acceptable, negative divided by positive gives
a negative answer, so we could also write
=−
1
36
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
24
37. Start by changing subtraction into addition.
8 4
8
4
− − =− + −
9 5
9
5
Add the opposite.
Make equivalent fractions over a common denominator. Then add the numerators over the common denominator and simplify.
4 9
8 5
=− · + − ·
9 5
5 9
36
40
=− + −
45
45
=
−40 + (−36)
45
=
−76
45
Make equivalent fractions
with LCD = 45.
Simplify numerators and
denominators.
Add numerators over common
denominator.
Simplify numerator.
Although this answer is perfectly acceptable, negative divided by positive gives
a negative answer, so we could also write
=−
76
45
39. Evaluate the expression inside the absolute value bars first.
8 5 2 8 5
2 − − = − + − 9
2 5
9
2
5
2 2 5 5
8
= − · + − ·
9
2 5
5 2 4 8 25
= − + −
9
10
10 8 21 = − 9
10
Second Edition: 2012-2013
Add the opposite.
Make equivalent fractions
with LCD = 10.
Add numerators over
Common denominator.
1.3. THE RATIONAL NUMBERS
25
Next, take the absolute value, then make equivalent fractions with a common
denominator and simplify.
=
=
=
=
=
8 21
−
9 10
8
21
+ −
9
10
21 9
8 10
·
+ − ·
9 10
10 9
189
80
+ −
90
90
−109
90
Take absolute value.
Add the opposite.
Make equivalent fractions
with LCD = 90.
Add.
41. First, evaluate the exponent, then multiply.
2 7
1
5
−
+ −
−
6
2
3
1
49
5
+ −
=
−
Evaluate exponent.
36
2
3
49 5
+
Multiply.
=
36 6
Make equivalent fractions with a common denominator, add numerators over
a common denominator and simplify.
49
36
49
=
36
79
=
36
=
1 5 6
+ ·
1 6 6
30
+
36
·
Make equivalent fractions
with LCD = 36.
Add over common denominator.
43. Multiply first, then add or subtract as needed.
8
9
9
1
−
−
+
−
5
7
5
2
4
81
+ −
=
Multiply.
35
5
4 7
81 1
· + − ·
=
Make equivalent fractions
35 1
5 7
28
81
+ −
with LCD = 35.
=
35
35
53
Add over common denominator.
=
35
Second Edition: 2012-2013
26
CHAPTER 1. THE ARITHMETIC OF NUMBERS
45. Multiply first, then add or subtract as
5 7
63
9
5
− +
−
=− + −
8 2
2
8
4
63 2
5 1
=− · + − ·
8 1
4 2
126
5
=− + −
8
8
131
=−
8
needed.
Multiply.
Make equivalent fractions
with LCD = 8.
Add over common denominator.
47. First evaluate exponent, then multiply.
2
7
9
2
−
− −
5
2
5
9
7
4
Exponent first.
= −
−
5
2
25
63
4
=− −
Multiply.
10 25
63
4
=− + −
Add the opposite.
10
25
Make equivalent fractions with a common denominator, then add numerators
over a common denominator and simplify.
4 2
63 5
Make equivalent fractions
=− · + − ·
10 5
25 2
8
315
+ −
=−
with LCD = 50.
50
50
323
Add over common denominator.
=−
50
49. Multiply first, then add or subtract
6 2
8
4
6
−
−
= − −
5 5
9
5
45
8
6
= +
5 45
6 9
8 1
= · +
·
5 9 45 1
54
8
=
+
45 45
62
=
45
Second Edition: 2012-2013
as needed.
Multiply.
Add the opposite.
Make equivalent fractions
with LCD = 45.
Add over common denominator.
1.3. THE RATIONAL NUMBERS
51. Multiply first.
2
4
8
9
−
−
−
3
7
7
8
9
16
=− − −
21
14
9
16
=− +
21 14
27
Multiply.
Add the opposite.
Next, make equivalent fractions with a common denominator, add numerators
over a common denominator and simplify.
16
21
32
=−
42
5
=−
42
=−
2
9 3
+
·
2 14 3
27
+
42
·
Make equivalent fractions
with LCD = 42.
Add over common denominator.
53. Replace each variable with open parentheses, then substitute −1/2 for x,
−1/3 for y, and 5/2 for z.
2
xy − z 2 = ( ) ( ) − ( )
2
5
1
1
= −
−
−
2
3
2
Replace variable with parentheses.
Substitute −1/2 for x, −1/3
for y, and 5/2 for z.
First evaluate exponent, then multiply.
1
1
25
= −
Exponent first.
−
−
2
3
4
1 25
= −
Multiply.
6 4
1
25
= + −
Add the opposite.
6
4
Make equivalent fractions with a common denominator, then add numerators
over a common denominator and simplify.
1 2
25 3
= · + − ·
Make equivalent fractions
6 2
4 3
75
2
+ −
with LCD = 12.
=
12
12
73
Add over common denominator.
=−
12
Second Edition: 2012-2013
28
CHAPTER 1. THE ARITHMETIC OF NUMBERS
55. Replace each variable with open parentheses, then substitute 3/4 for x
and −1/2 for y.
2
2
−5x2 + 2y 2 = −5 ( ) + 2 ( )
2
2
3
1
= −5
+2 −
4
2
Exponents first: (3/4)2 = 9/16 and (−1/2)2 = 1/4.
= −5
9
16
1
+2
4
Multiply: −5(9/16) = −45/16 and 2(1/4) = 1/2
=−
45 1
+
16 2
Make equivalent fractions with LCD = 16, then add the numerators over the
common denominator and simplify.
45
16
45
=−
16
−37
=
16
=−
1 1 8
+ ·
1 2 8
8
+
16
·
57. Replace each variable with open parentheses, then substitute 3/2 for x
and −3/4 for y.
2
2
2x2 − 2xy − 3y 2 = 2 ( ) − 2 ( ) ( ) − 3 ( )
2
2
3
3
3
3
=2
−2
−
−3 −
2
2
4
4
Exponents first: (3/2)2 = 9/4 and (−3/4)2 = 9/16.
=2
9
3
9
3
−2
−
−3
4
2
4
16
Multiply: 2(9/4) = 9/2, 2(3/2)(−3/4) = −9/4, and 3(9/16) = 27/16.
=
Second Edition: 2012-2013
9
9
27
− −
−
2
4
16
1.3. THE RATIONAL NUMBERS
29
Add the opposite.
9 9
27
= + + −
2 4
16
Make equivalent fractions with LCD = 16, then add the numerators over the
common denominator and simplify.
27 1
9 8 9 4
= · + · + − ·
2 8 4 4
16 1
27
72 36
+
+ −
=
16 16
16
81
=
16
59. Replace all variables with open parentheses, then substitute −1/3 for x,
1/6 for y, and 2/5 for z.
x + yz = ( ) + ( ) ( )
1
1
2
= −
−
3
6
5
Replace variables with parentheses.
Substitute −1/3 for x, 1/6
for y, and 2/5 for z.
Multiply first, then subtract.
1
1
=− −
3 15
1
1
=− + −
3
15
Multiply.
Add the opposite.
Make equivalent fractions with a common denominator, then add numerators
over the common denominator and simplify.
1 1
1 5
=− · + − ·
3 5
15 1
1
5
=− + −
15
15
6
=−
15
2
=−
5
Make equivalent fractions
with LCD = 15.
Add over common denominator.
Simplify.
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
30
61. Replace each variable with open parentheses, then substitute 4/7 for a,
7/5 for b, and 5/2 for c.
ab + bc = ( ) ( ) + ( ) ( )
4
7
7
5
= −
+
−
7
5
5
2
Multiply and reduce.
28
35
+ −
35
10
7
4
=− + −
5
2
=−
Make equivalent fractions with LCD(5, 2) = 10, then add numerators over a
common denominator and simplify.
4 2
7 5
=− · + − ·
5 2
2 5
35
8
=− + −
10
10
−8 + (−35)
=
10
43
=−
10
63. Replace each occurrence of the variable x with open parentheses, then
substitute −1/2 for x.
x3 = ( )3
3
1
= −
2
Replace variable with
open parentheses.
Substitute −1/2 for x.
In the expression (−1/2)3 , the exponent 3 tells us to write the base −1/2 three
times as a factor.
1
1
1
= −
−
−
Write −1/2 as a
2
2
2
factor three times.
1
The product of three
=−
8
negative fractions is
negative.
Second Edition: 2012-2013
1.3. THE RATIONAL NUMBERS
31
65. Replace all variables with open parentheses, then substitute −8/5 for x,
1/3 for y, and −8/5 for z.
x − yz = ( ) − ( ) ( )
8
8
1
= −
−
−
5
3
5
Replace variables with parentheses.
Substitute −8/5 for x, 1/3
for y, and −8/5 for z.
Multiply first, then add or subtract as needed.
8
8
=− − −
Multiply.
5
15
8
8
Add the opposite.
=− +
5 15
Make equivalent fractions with a common denominator, then add numerators
over the common denominator and simplify.
8 3
8 1
=− · +
·
5 3 15 1
24
8
=− +
15 15
16
=−
15
Make equivalent fractions
with LCD = 15.
Add over common denominator.
67. Replace each occurrence of the variable x with open parentheses, then
substitute −8/3 for x. Note that we must deal with the exponent first, then
negate our answer in the final step.
2
−x2 = − ( )
2
8
=− −
3
Replace variable with
open parentheses.
Substitute −8/3 for x.
The exponent 2 tells us to write the base −8/3 two times as a factor.
8
8
=− −
−
3
3
64
=−
9
=−
64
9
Write −8/3 as a
factor two times.
The product of two
negative fractions is
positive.
Negate.
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
32
69. First, replace each variable with open parentheses, then substitute 7/2 for
x, −5/4 for y, and −5/3 for z.
2
x2 + yz = ( ) + ( ) ( )
2 7
5
5
=
+ −
−
2
4
3
Replace variables with parentheses.
Substitute: 7/2 for x, −5/4
for y, and −5/3 for z
Next, evaluate the exponent, then multiply.
5
49
5
+ −
=
−
Evaluate exponent.
4
4
3
49 25
+
Multiply.
=
4
12
Make equivalent fractions with a common denominator, add numerators over
a common denominator and simplify.
49 3 25 1
· +
·
4 3 12 1
147 25
=
+
12
12
172
=
12
43
=
3
=
Make equivalent fractions
with LCD = 12.
Add over common denominator.
Simplify.
71. The Rules Guiding Order of Operations requires that we first perform the
division. Hence:
b
a + b/c + d = a + + d
c
73. The Rules Guiding Order of Operations requires that we attend the parentheses first, then the division, then the addition. Hence:
a + b/(c + d) = a +
b
c+d
75. Enter the expression 4125/1155, then press the ENTER key. Select
1:Frac from the MATH menu, then press ENTER again. The result is shown
in the following figure.
Second Edition: 2012-2013