242
Chapter 9
◆
Fractions and Fractional Equations
Mixed Form
Rewrite each mixed expression as an improper fraction.
1
57. x x
1
58. x
2
x
1
59. 1 x1
x1
2
61. 3 a a
1
60. a 2
a b2
5
3
62. 2x 2
x
x
2x 3
x2
64. p 5x q p 2x q
3
4
63.
p
bq
1q p
3b 2b x
ax
Applications
Treat the given numbers in these problems as exact, and leave your answers in fractional form.
Hint: In problem 65, first find the
amount that each crew can do in
one day (e.g., the first crew can
grade 7/3 km per day). Then add
the separate amounts to get the
daily total. You can use a similar
approach to the other work
problems in this group.
Do this group of problems without
using your calculator, and leave
your answers in fractional form.
a
h
65. A certain work crew can grade 7 km of roadbed in 3 days, and another crew can do 9 km
in 4 days. How much can both crews together grade in 1 day?
66. Liquid is running into a tank from a pipe that can fill four tanks in 3 days. Meanwhile,
liquid is running out from a drain that can empty two tanks in 4 days. What will be the net
change in volume in 1 day?
67. A planer makes a 1 m cutting stroke at a rate of 15 m/min and returns at 75 m/min. How
long does it take for the cutting stroke and return? (See Eq. A17.)
68. Two resistors, 5 and 15, are wired in parallel. What is the resistance of the parallel
combination? (See Eq. A64.)
69. One crew can put together five machines in 8 days. Another crew can assemble three of these
machines in 4 days. How much can both crews together assemble in 1 day?
70. One crew can assemble M machines in p days. Another crew can assemble N of these
machines in q days. Write an expression for the number of machines that both crews
together can assemble in 1 day, combine into a single term, and simplify.
71. A steel plate in the shape of a trapezoid (Fig. 9–3) has a hole of diameter d. The
area of the plate, less the hole, is
(a b)h d 2
2
4
d
Combine these two terms and simplify.
72. The total resistance R of Fig. 9–4 is
b
FIGURE 9–3
R1R2
R3
R1 R2
R1
R3
Combine into a single term and simplify.
73. If a car travels a distance d at a rate V, the time required will be d/V. The car then
continues for a distance d1 at a rate V1, and a third distance d2 at rate V2. Write an
expression for the total travel time; then combine the three terms into a single term
and simplify.
R2
R
FIGURE 9–4
9–4
Complex Fractions
Fractions that have only one fraction line are called simple fractions. Fractions with more than
one fraction line are called complex fractions.
Section 9–4
◆◆◆
◆
243
Complex Fractions
Example 32: The following are complex fractions:
a
1
x
b
x
and c
x
x
y
We show how to simplify complex fractions in the following examples.
◆◆◆
◆◆◆
Example 33: Simplify the complex fraction
1 2
2 3
1
3
4
Solution: We can simplify this fraction by multiplying numerator and denominator by the least
common denominator for all of the individual fractions. The denominators are 2, 3, and 4, so
the LCD is 12. Multiplying, we obtain
p
2q
3
68
14
39
36
3
1
p
3 q12
4
◆◆◆
1
2
12
◆◆◆
Example 34: Simplify the complex fraction
a
1
b
b
1
a
Solution: The LCD for the two small fractions a/b and b/a is ab. Multiplying, we obtain
a
1
b ab ab a2
• b ab ab b2
1
a
or, in factored form,
a(b a)
b(a b)
◆◆◆
In the following example, we simplify a small section at a time and work outward. Try to
follow the steps.
◆◆◆
Example 35:
ab 1
1
a
b
ab
b
ab
• • 1
1
ab 1
ab 1
a a 1
ab 1
b
a
a
ab 1
b
ab
• a
a ab 1
ab 1
ab 1
b
ab
• a(ab 1) a ab 1
ab 1
244
Chapter 9
◆
Fractions and Fractional Equations
ab 1
ab 1
ab
• • b
a(ab 1) a ab 1
a(ab 1)
ab 1
a(ab 1) a ab 2
Exercise 4
◆
Complex Fractions
Simplify. Leave your answers as improper fractions.
2
3
3
4
1. 1
5
1 1 1
2 3 4
3. 4
3
5
2
5
5
5. 1
6
3
y
x
4
7. y
x
3
x
1
y
9. 2
x
1 2
y
x
a2 3
11. x
4
5
2d
x 3ac
13. 3d
x 2ac
y2
x2 2
15. x 3y
2
1
1 x1
17. 1
1 x1
2m n
1
mn
19. n
1 mn
3 1
4 3
2. 1 1
2 6
4
5
4. 1 2
5 3
1
3
6. 2 2 1
5 3
a x
b y
8. a x
z c
a
x
c
10. b
x
d
3a2 3y2
12. ay
3
4a2 4x2
14. ax
ax
ab
7
3d
16. ab
3c d
3x
xy ac
18. ac
2c
x
x3 y3
x2 y2
20. x2 xy y2
xy
◆◆◆
Section 9–5
◆
245
Fractional Equations
xy
22. 1
x y 1
x y xy
1
21. 1
a a1
1 3a
a1
a
b1 c1
b
c
3abc
23. bc ac ab
1 1 1
a b c
1
1
a bcp
b2 c2 a2 q
24. 1 1
1
2bc
a bc
x2 y2 z2 2yz
25. x2 y2 z2 2yz
xyz
xyz
Applications
26. A car travels a distance d1 at a rate V1, then another distance d2 at a rate V2. The average
speed for the entire trip is
d1 d2
average speed d1
d2
V1 V2
Simplify this complex fraction.
27. The equivalent resistance of two resistors in parallel is
R1R2
R1 R2
If each resistor is made of wire of resistivity , with R1 using a wire of length L1 and crosssectional area A1, and R2 having a length L2 and area A2, our expression becomes
L1 L2
• A1 A2
L1
L2
A1
A2
Simplify this complex fraction.
9–5
Fractional Equations
Solving Fractional Equations
An equation in which one or more terms is a fraction is called a fractional equation. To solve
a fractional equation, first eliminate the fractions by multiplying both sides of the equation
by the least common denominator (LCD) of every term. We can do this because multiplying
See Eq. A71, R L/A.