372
(6-40)
Chapter 6
Rational Expressions
67. Wealth-building portfolio. Misty decided to invest her annual bonus in a wealth-building portfolio as shown in the
figure (Fidelity Investments, Boston).
a) If the amount that she invested in stocks was $20,000
greater than her investment in bonds, then how much
did she invest in bonds?
b) What was the amount of her annual bonus?
a) $17,142.86 b) $57,142.86
Designing a retirement portfolio
Captial preservation
portfolio
Moderate
portfolio
formula N (1 1C)B 1 to estimate the total number
of such weapons N that the enemy has produced (New Scientist, May 1998). B is the biggest serial number obtained
and C is the number of weapons obtained. It is assumed the
weapons are numbered 1 through N.
a) Find N if agents obtain five nerve gas containers numbered 45, 143, 258, 301, and 465.
b) Find C if agents estimate that the enemy has 255 tanks
from a group of captured tanks on which the biggest
serial number is 224.
a) 557 b) 7
GET TING MORE INVOLVED
50%
20%
20%
40%
30%
40%
69. Writing. In this chapter the LCD is used to add rational
expressions and to solve equations. Explain the difference
between using the LCD to solve the equation
3
7
2
x2 x2
Wealth-building
portfolio
and using the LCD to find the sum
3
7
.
x2 x2
Short term
65%
5%
30%
Bonds
Stocks
FIGURE FOR EXERCISE 67
68. Estimating weapons. When intelligence agents obtain
enemy weapons marked with serial numbers, they use the
6.6
In this
section
●
Formulas
●
Uniform Motion Problems
●
Work Problems
●
Miscellaneous Problems
E X A M P L E
70. Discussion. For each equation, find the values for x that
cannot be solutions to the equation. Do not solve the
equations.
1
x
1
1
1
a) b) x1 2
x x1 2
1
1
c) 2
x 1 x1
a) 0, 1 b) 1 c) 1
APPLICATIONS
In this section we will use the techniques of Section 6.5 to rewrite formulas involving rational expressions and to solve some problems.
Formulas
Rewriting formulas having rational expressions is similar to solving equations
having rational expressions. Generally, the first step is to multiply each side by the
LCD for the rational expressions.
1
Solving a formula
In Chapter 3, we wrote the equation of a line by starting with an equation involving
a rational expression:
y y1
m
x x1
Solve the equation for y.
6.6
Applications
(6-41)
373
Solution
helpful
hint
When this equation was written in the form
yy
(x x1)1 (x x1)m
x x1
y y1 (x x1)m
Reduce.
y (x x1)m y1
y y1 m(x x1)
in Chapter 3, we called it
the point-slope formula for
the equation of a line.
E X A M P L E
Multiply each side by the denominator x x1.
2
■
In the next example we solve for a variable that occurs twice in the original formula. Remember that when a formula is solved for a certain variable, that variable
must appear only once in the final formula.
Solving for a variable
The formula
P
2L
PW 2L d
is used in physics to find the relative density of a substance. Since PW has subscript
W, we treat P and PW as two different variables. Solve the formula for L.
Solution
study
P
2L
PW 2L d
tip
As you study from the text,
think about the material. Ask
yourself questions. If you were
the professor, what questions
would you ask on the test?
P(2L d ) PW (2L)
The extremes-means property
2PL Pd 2LPW
Simplify.
Pd 2LPW 2PL
Get all terms involving L onto the same side.
Pd (2PW 2P)L
Factor out L.
Pd
L
2PW 2P
■
In the next example we find the value of one variable when given the values of
the remaining variables.
E X A M P L E
3
Evaluating a formula
1
Find x if x1 2, y1 3, y 1, m 2, and
y y1
m.
x x1
Solution
Substitute all of the values into the formula and solve for x:
1 (3) 1
x2
2
2
1
x2 2
x24
x6
Substitute.
Extremes-means property
Check in the original formula.
■
374
(6-42)
Chapter 6
Rational Expressions
Uniform Motion Problems
The uniform motion problems here are similar to those of Chapter 2, but in this
chapter the equations have rational expressions.
E X A M P L E
300 mi
Speed = x mph
300 mi
Speed = x – 10 mph
4
Uniform motion
Michele drove her empty rig 300 miles to Salina to pick up a load of cattle. When
her rig was fully loaded, her average speed was 10 miles per hour less than when the
rig was empty. If the return trip took her 1 hour longer, then what was her average
speed with the rig empty? (See Fig. 6.1.)
Solution
Let x be Michele’s average speed empty and let x 10 be her average speed full.
D
Because the time can be determined from the distance and the rate, T R, we can
make the following table.
FIGURE 6.1
Rate
Time
Empty
i
xm
300
x
Full
i
x 10 m
hr
hr
Distance
hr
300
x 10
300 mi
hr
300 mi
We now write an equation expressing the fact that her time empty was 1 hour less
than her time full:
300
300
1
x
x 10
300
300
x(x 10) x(x 10) x(x 10)1
x
x 10
300x 3000 300x x 2 10x
3000 x 2 10x
2
x 10x 3000 0
(x 50)(x 60) 0
x 50 0
or
x 60 0
x 50
or
x 60
Multiply each side by
x(x 10).
Reduce.
Get 0 on one side.
Factor.
Zero factor property
The equation is satisfied if x 50, but because 50 is negative, it cannot be the
speed of the truck. Michele’s average speed empty was 60 miles per hour (mph).
Checking this answer, we find that if she traveled 300 miles at 60 mph, it would take
her 5 hours. If she traveled 300 miles at 50 mph with the loaded rig, it would take
her 6 hours. Because Michele’s time with the empty rig was 1 hour less than her
■
time with the loaded rig, 60 mph is the correct answer.
Work Problems
Problems involving different rates for completing a task are referred to as work
problems. We did not solve work problems earlier because they usually require
equations with rational expressions. Work problems are similar to uniform motion
problems in which RT D. The product of a person’s time and rate is the amount
6.6
(6-43)
Applications
375
of work completed. For example, if your puppy gains 1 pound every 3 days, then he
is growing at the rate of 13 pound per day. If he grows at the rate of 13 pound per day for
a period of 30 days, then he gains 10 pounds.
E X A M P L E
helpful
5
hint
The secret to work problems is
remembering that the individual amounts of work or the individual rates can be added
when people work together. If
your painting rate is 1 of the
10
house per day and your
1
helper’s rate is of the house
5
per day, then your rate together will be 3 of the house
10
per day.
Working together
Linda can mow a certain lawn with her riding lawn mower in 4 hours. When Linda
uses the riding mower and Rebecca operates the push mower, it takes them 3 hours
to mow the lawn. How long would it take Rebecca to mow the lawn by herself using
the push mower?
Solution
If x is the number of hours it takes for Rebecca to complete the lawn alone, then her
rate is 1 of the lawn per hour. Because Linda can mow the entire lawn in 4 hours,
x
her rate is 1 of the lawn per hour. In the 3 hours that they work together, Rebecca com4
pletes 3 of the lawn while Linda completes 3 of the lawn. We can classify all of the
x
4
necessary information in a table that looks a lot like the one we used in Example 4.
Rate
Time
Amount of Work
Linda
1 lawn
4 hr
3 hr
3
4
lawn
Rebecca
1 lawn
x hr
3 hr
3
x
lawn
Because the lawn is finished in 3 hours, the two portions of the lawn (in the work
column) mowed by each girl have a sum of 1:
3 3
1
4 x
3
3
4x 4x 4x 1
4
x
3x 12 4x
12 x
Multiply each side by 4x.
If x 12, then in the 3 hours that they work together, Rebecca does 3 or 1 of the job
12
4
while Linda does 3 of the job. So it would take Rebecca 12 hours to mow the lawn
4
by herself using the push mower.
■
Miscellaneous Problems
E X A M P L E
6
Hamburger and steak
Patrick bought 50 pounds of meat consisting of hamburger and steak. Steak costs
twice as much per pound as hamburger. If he bought $30 worth of hamburger and
$90 worth of steak, then how many pounds of each did he buy?
Solution
Let x be the number of pounds of hamburger and 50 x be the number of pounds
of steak. Because Patrick got x pounds of hamburger for $30, he paid 30 dollars per
x
pound for the hamburger. We can classify all of the given information in a table.
376
(6-44)
Chapter 6
Rational Expressions
Price per pound
Amount
Total price
30 dollars
x
lb
x lb
30 dollars
50 x lb
90 dollars
Hamburger
90 dollars
50 x lb
Steak
Because the price per pound of steak is twice that of hamburger, we can write the
following equation:
30
90
2 x
50 x
90
60
50 x
x
90x 3000 60x
150x 3000
x 20
50 x 30
The extremes-means property
Patrick purchased 20 pounds of hamburger and 30 pounds of steak. Check this
■
answer.
WARM-UPS
True or false? Explain.
1. The formula w 1t, solved for t, is t 1t. False
t
w
1
p
1
q
1
s
2. To solve for s, multiply each side by pqs.
True
50
3. If 50 pounds of steak cost x dollars, then the price is dollars per pound.
x
False
x
4. If Claudia drives x miles in 3 hours, then her rate is miles per hour. True
3
1
5. If Takenori mows his entire lawn in x 2 hours, then he mows of the
x2
lawn per hour. True
6. If Kareem drives 200 nails in 12 hours, then he is driving 200 nails per hour.
12
True
7. If x hours is 1 hour less than y hours, then x 1 y. False
AB
v2
2
8. If A m
and m and B are nonzero, then v . True
B
x
y
m
9. If a and y are nonzero and a , then y ax. False
10. If x hours is 3 hours more than y hours, then x 3 y. False
6. 6
EXERCISES
Solve each equation for y. See Example 1.
y3
1. 5 y 5x 7
x2
y4
2. 6 y 6x 46
x7
y1
1
1
3. y x 1
x6
3
3
y 7 2
2
17
4. y x x2
3
3
3
ya
5. m
xb
yh
6. a
xk
y mx bm a
y ax ak h
6.6
y2
7
7
29
7. y x x5
3
3
3
y3
9
9
3
8. y x x1
4
4
4
Solve each formula for the indicated variable. See Example 2.
F
A
9. M for f
10. P for A
f
1 rt
F
A P(1 rt)
f M
12. V r 2h for r 2
11. A D 2 for D 2
4
4A
V
D2 r 2 h
m1m2
mv 2
13. F k for m1
14. F for v 2
r2
r
Fr 2
rF
2
m 1 v km2
m
1 1 1
1
1
1
15. for q
16. for R1
p q
f
R R1 R2
RR2
pf
q R1 pf
R2 R
b2
b2
2
2
2
17. e 1 2 for a
18. e 1 2 for b 2
a
a
b2
2
2
2
a 2
b a a2e2
1e
PV
PV
PV
PV
19. 11 22 for T1
20. 11 22 for P2
T1
T2
T1
T2
P1V1T2
P1V1T2
T1 P2 P2V2
T1V2
4 2
S 2r 2
21. V r h for h
22. h for S
3
2r
3V
S 2rh 2r 2
h 2
4r
Use the formula from the indicated exercise to find the value of
the indicated variable. See Example 3. For calculator problems,
round answers to three decimal places.
1
23. If M 10 and F 5 in Exercise 9, find f. 2
24. If A 550, P 500, and t 2 in Exercise 10, find r.
0.05
25. If A 6 in Exercise 11, find D2. 24
4
26. If V 12 and r 3 in Exercise 12, find h. 3
27. If F 32, r 4, m1 6, and m 2 8 in Exercise 13,
32
find k. 3
144
28. If F 10, m 8, and v 6 in Exercise 14, find r. 5
29. If f 2.3 and q 1.7 in Exercise 15, find p. 6.517
30. If R 1.29 and R1 0.045 in Exercise 16, find R2.
0.046
31. If e 0.62 and b 3.5 in Exercise 17, find a2. 19.899
Applications
(6-45)
377
32. If a 3.61 and e 2.4 in Exercise 18, find b2. 62.033
33. If V 25.6 and h 3.2 in Exercise 21, find r 2. 1.910
34. If h 3.6 and r 2.45 in Exercise 22, find S. 93.133
Solve each problem. See Examples 4–6.
35. Walking and riding. Karen can ride her bike from home to
school in the same amount of time as she can walk from
home to the post office. She rides 10 miles per hour (mph)
faster than she walks. The distance from her home to school
is 7 miles, and the distance from her home to the post office
is 2 miles. How fast does Karen walk? 4 mph
Grant Hall
P.O.
7 mi
2 mi
FIGURE FOR EXERCISE 35
36. Fast driving. Beverly can drive 600 miles in the same time
as it takes Susan to drive 500 miles. If Beverly drives
10 mph faster than Susan, then how fast does Beverly
drive? 60 mph
37. Faster driving. Patrick drives 40 miles to work, and Guy
drives 60 miles to work. Guy claims that he drives at the
same speed as Patrick, but it takes him only 12 minutes
longer to get to work. If this is true, then how long does it
take each of them to get to work? What are their speeds?
Patrick 24 minutes; Guy 36 minutes, 100 mph
38. Route drivers. David and Keith are route drivers for a fastphoto company. David’s route is 80 miles, and Keith’s is
100 miles. Keith averages 10 mph more than David and finishes his route 10 minutes before David. What is David’s
speed? 30 mph
39. Physically fit. Every morning, Yong Yi runs 5 miles, then
walks one mile. He runs 6 mph faster than he walks. If his
total time yesterday was 45 minutes, then how fast did he
run? 10 mph
40. Row, row, row your boat. Norma can row her boat
12 miles in the same time as it takes Marietta to cover 36
miles in her motorboat. If Marietta’s boat travels 15 mph
faster than Norma’s boat, then how fast is Norma rowing
her boat? 7.5 mph
41. Pumping out the pool. A large pump can drain an
80,000-gallon pool in 3 hours. With a smaller pump also
operating, the job takes only 2 hours. How long would
it take the smaller pump to drain the pool by itself?
6 hours
(6-46)
Chapter 6
Rational Expressions
42. Trimming hedges. Lourdes can trim the hedges around her
property in 8 hours by using an electric hedge trimmer.
Rafael can do the same job in 15 hours by using a manual
trimmer. How long would it take them to trim the hedges
working together?
120
hours
23
43. Filling the tub. It takes 10 minutes to fill Alisha’s bathtub
and 12 minutes to drain the water out. How long would it
take to fill it with the drain accidentally left open?
60 minutes
FIGURE FOR EXERCISE 43
44. Eating machine. Charles can empty the cookie jar in 11
2
hours. It takes his mother 2 hours to bake enough cookies to
fill it. If the cookie jar is full when Charles comes home
from school, and his mother continues baking and restocking the cookie jar, then how long will it take him to empty
the cookie jar?
6 hours
45. Filing the invoices. It takes Gina 90 minutes to file the
monthly invoices. If Hilda files twice as fast as Gina does,
how long will it take them working together?
30 minutes
46. Painting alone. Julie can paint a fence by herself in
12 hours. With Betsy’s help, it takes only 5 hours. How
long would it take Betsy by herself?
60
hours
7
47. Buying fruit. Molly bought $5.28 worth of oranges and
$8.80 worth of apples. She bought 2 more pounds of oranges than apples. If apples cost twice as much per pound
as oranges, then how many pounds of each did she buy?
10 pounds apples, 12 pounds oranges
48. Raising rabbits. Luke raises rabbits and raccoons to sell
for meat. The price of raccoon meat is three times the price
of rabbit meat. One day Luke sold 160 pounds of meat,
$72 worth of each type. What is the price per pound of each
type of meat?
Rabbit $0.60 per pound, raccoon $1.80 per pound
49. Total resistance. If two receivers with resistances R1 and
R2 are connected in parallel, then the formula
1
1
1
R R1 R2
relates the total resistance for the circuit R with R1 and R2.
Given that R1 is 3 ohms and R is 2 ohms, find R2.
6 ohms
R1
R2
FIGURE FOR EXERCISE 49
50. More resistance. Use the formula from Exercise 49 to find
R1 and R2 given that the total resistance is 1.2 ohms and R1
is 1 ohm larger than R2.
R2 2 ohms, R1 3 ohms
51. Las Vegas vacation. Brenda of Horizon Travel has arranged for a group of gamblers to share the $24,000 cost of
a charter flight to Las Vegas. If Brenda can get 40 more people to share the cost, then the cost per person will decrease
by $100.
a) How many people were in the original group?
b) Write the cost per person as a function of the number of
people sharing the cost.
24,000
a) 80
b) C(n) n
Cost per person (in dollars)
378
500
400
300
200
100
0
0
100
200
300
Number of people
FIGURE FOR EXERCISE 51
52. White-water rafting. Adventures, Inc. has a $1,500 group
rate for an overnight rafting trip on the Colorado River. For
the last trip five people failed to show, causing the price per
person to increase by $25. How many were originally
scheduled for the trip?
20
53. Doggie bag. Muffy can eat a 25-pound bag of dog food in
28 days, whereas Missy eats a 25-pound bag in 23 days.
How many days would it take them together to finish a
50-pound bag of dog food.
25.255 days
54. Rodent food. A pest control specialist has found that 6 rats
can eat an entire box of sugar-coated breakfast cereal in
13.6 minutes, and it takes a dozen mice 34.7 minutes to devour the same size box of cereal. How long would it take all
18 rodents, in a cooperative manner, to finish off a box of
cereal?
9.7706 minutes