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Section 6.6 Rational Equations and Problem Solving 381
PRACTICE
OBJECTIVE
1 Solving Equations with Rational Expressions
for a Specified Variable
In Section 2.3, we solved equations for a specified variable. In this section, we continue practicing this skill by solving equations containing rational expressions for a specified variable. The steps given in Section 2.3 for solving equations for a specified variable are repeated here.
OBJECTIVES
1 Solve an Equation Containing Rational Expressions for a Specified Variable. 2 Solve Number Problems by Writing Equations Containing Rational Expressions. 3 Solve Problems Modeled by Proportions. 4 Solve Problems About Work. 5 Solve Problems About Distance, Rate, and Time.
6.6 Rational Equations and Problem Solving
Solving Equations for a Specified Variable Step 1. Clear the equation of fractions or rational expressions by multiplying each side of the equation by the least common denominator (LCD) of all denominators in the equation. Step 2. Use the distributive property to remove grouping symbols such as parentheses. Step 3. Combine like terms on each side of the equation. Step 4. Use the addition property of equality to rewrite the equation as an equivalent equation with terms containing the specified variable on one side and all other terms on the other side. Step 5. Use the distributive property and the multiplication property of equality to get the specified variable alone.
E X A M P L E 1 Solve:
x
y =^
z for x.
Solution To clear this equation of fractions, we multiply both sides of the equation by
xyz , the LCD of
x
y , and
z
x
y =^
z
xyz a
x
y b^ =^ xyz a^
z b
xyz a
x b + xyz a
y b^ =^ xyz a^
z b
yz + xz = xy
Multiply both sides by xyz.
Use the distributive property.
Simplify. Notice the two terms that contain the specified variable x. Next, we subtract xz from both sides so that all terms containing the specified vari- able x are on one side of the equation and all other terms are on the other side. yz = xy - xz Now we use the distributive property to factor x from xy - xz and then the multipli- cation property of equality to solve for x. yz = x 1 y - z 2 yz y - z =^ x^ or^ x^ =^
yz y - z Divide both^ sides^ by^ y^ -^ z.
1 Solve:
a
b
c for a.
382 CHAPTER 6 Rational Expressions
PRACTICE
T T T
OBJECTIVE
2 Solving Number Problems Modeled by Rational Equations
Problem solving sometimes involves modeling a described situation with an equation containing rational expressions. In Examples 2 through 5, we practice solving such problems and use the problem-solving steps first introduced in Section 2.2.
E X A M P L E 2 Finding an Unknown Number
If a certain number is subtracted from the numerator and added to the denominator of 9 19 , the new fraction is equivalent to
. Find the number.
Solution
1. UNDERSTAND the problem. Read and reread the problem and try guessing the solution. For example, if the unknown number is 3, we have 9 - 3 19 + 3
To see if this is a true statement, we simplify the fraction on the left side. 6 22
or
False Since this is not a true statement, 3 is not the correct number. Remember that the purpose of this step is not to guess the correct solution but to gain an understanding of the problem posed. We will let n = the number to be subtracted from the numerator and added to the denominator.
2. TRANSLATE the problem.
In words:
when the number is subtracted from the numerator and added to the denominator of the fraction
this is equivalent to
Translate: 9 - n 19 + n
3. SOLVE the equation for n. 9 - n 19 + n
To solve for n , we begin by multiplying both sides by the LCD of 3 119 + n 2.
3119 + n 2 #^ 9 - n 19 + n = 3119 + n 2 #^
319 - n 2 = 19 + n 27 - 3 n = 19 + n 8 = 4 n 2 = n
Multiply both sides by the LCD. Simplify.
Solve.
4. INTERPRET the results. Check: If we subtract 2 from the numerator and add 2 to the denominator of
, we have
, and the problem checks. State: The unknown number is 2.
2 Find a number that when added to the numerator and subtracted from the denominator of
results in a fraction equivalent to
384 CHAPTER 6 Rational Expressions
T T T T T
Translate:
t
OBJECTIVE
PRACTICE
4. INTERPRET.
Check: To check, replace x with 10,080 in the proportion and see that a true statement results. Notice that our answer is reasonable since it is less than 12,000 as we stated in Step 1. State: We predict that 10,080 homes are heated by electricity.
3 In the United States, 1 out of 12 homes is heated by fuel oil. At this rate, how many homes in a community of 36,000 homes are heated by fuel oil? ( Source: American Housing Survey for the United States )
4 Solving Problems About Work
The following work example leads to an equation containing rational expressions.
E X A M P L E 4 Calculating Work Hours
Melissa Scarlatti can clean the house in 4 hours, whereas her husband, Zack, can do the same job in 5 hours. They have agreed to clean together so that they can finish in time to watch a movie on TV that starts in 2 hours. How long will it take them to clean the house together? Can they finish before the movie starts?
Solution
1. UNDERSTAND. Read and reread the problem. The key idea here is the relation- ship between the time (in hours) it takes to complete the job and the part of the job completed in 1 unit of time (1 hour). For example, if the time it takes Melissa to complete the job is 4 hours, the part of the job she can complete in 1 hour is
Similarly, Zack can complete
of the job in 1 hour.
We will let t = the time in hours it takes Melissa and Zack to clean the house together. Then
t represents the part of the job they complete in 1 hour. We summarize the given information in a chart.
Hours to Complete the Job
Part of Job Completed in 1 Hour
MELISSA ALONE 4 1 4 ZACK ALONE 5 1 5 TOGETHER t 1 t
2. TRANSLATE.
In words:
part of job Melissa can complete in 1 hour
added to
part of job Zack can complete in 1 hour
is equal to
part of job they can complete together in 1 hour
Section 6.6 Rational Equations and Problem Solving 385
OBJECTIVE
PRACTICE
3. SOLVE.
t
20 t a
b = 20 t a
t b
5 t + 4 t = 20 9 t = 20
t =
or 2
Multiply both sides by the LCD, 20 t.
Solve.
4. INTERPRET.
Check: The proposed solution is 2
. That is, Melissa and Zack would take 2
hours
to clean the house together. This proposed solution is reasonable since 2
hours is
more than half of Melissa’s time and less than half of Zack’s time. Check this solution in the originally stated problem.
State: Melissa and Zack can clean the house together in 2
hours. They cannot complete the job before the movie starts.
5 Solving Problems About Distance, Rate, and Time
Before we solve Example 5, let’s review what we learned in Chapter 2 about the formula
d = r #^ t , or distance = rate #^ time
For example, if we travel at a rate or speed of 60 mph for a time of 3 hours, the distance we travel is
d = 60 mph #^ 3 hr = 180 mi
The formula d = r #^ t is solved for distance, d. We can also solve this formula for rate r or for time t.
4 Elissa Juarez can clean the animal cages at the animal shelter where she volunteers in 3 hours. Bill Stiles can do the same job in 2 hours. How long would it take them to clean the cages if they work together?
Solve d = r #^ t for r.
d = r #^ t d t
r #^ t t d t = r
Solve d = r #^ t for t. d = r #^ t d r
r #^ t r d r =^ t
All three forms of the distance formula are useful, as we shall see.
Solution
1. UNDERSTAND. Read and reread the problem. Guess a solution. Suppose that the current is 4 mph. The speed of the boat upstream is slowed down by the current:
E X A M P L E 5 Finding the Speed of a Current
Steve Deitmer takes 1
times as long to go 72 miles upstream in his boat as he does
to return. If the boat cruises at 30 mph in still water, what is the speed of the current?
(Continued on next page)
Section 6.6 Rational Equations and Problem Solving 387
PRACTICE 5 A tugboat takes 1
times as long to go 100 miles upstream along the Mississippi River as it does to return. If the speed of the current of the Mississippi River is 2 miles per hour, find the speed of the tugboat in still water.
Vocabulary, Readiness & Video Check
Martin-Gay Interactive Videos
See Video 6.
OBJECTIVE 1
OBJECTIVE 2
OBJECTIVE 3 OBJECTIVE 4
OBJECTIVE 5
Watch the section lecture video and answer the following questions.
1. In Example 1, an equation is solved for a specified variable. What is done differently here to get the specified variable alone on one side of the equation than was done in the past? 2. In general, if x represents a nonzero number, how do we represent its reciprocal? 3. For Example 3, how are units used to write a correct proportion? 4. From Example 4, how can you determine a somewhat reasonable answer to a work application before you even begin to solve it? 5. For Example 5, if the boat travels upstream at a rate of x - y , what is the boat’s rate downstream? Use the letters x and y.
6.6 Exercise Set
Solve each equation for the specified variable. See Example 1.
1. F = 95 C + 32 for C 2. V = 1 3 p r
(^2) h for h
3. Q = A^ -^ I L for I 4. P = 1 - CS for S 5.^1 R = 1 R 1 + 1 R 2 for R 6. (^) R^1 = (^) R^1 1
for R 1
7. S = n 1 a + L 2 2 for^ n 8. S =
n 1 a + L 2 2 for^ a
9. A = h 1 a + b 2 2 for b 10. A = h 1 a + b 2 2 for^ h
11. P 1 V 1 T 1 =^
P 2 V 2 T 2 for^ T^2
12. H = kA 1 T 1 - T 22 L for T 2 13. f = f 1 f 2 f 1 + f 2 for^ f^2 14. I = E R + r for r 15. l = (^2) nL for L 16. S = a 1 - a (^) nr 1 - r for a 1 17. (^) vu = (^2) cL for c 18. F = - GMm r^2
for M
Solve. For Exercises 19 and 20, the solutions have been started for you. See Example 2.
19. The sum of a number and 5 times its reciprocal is 6. Find the number(s). Start the solution: 1. UNDERSTAND the problem. Reread it as many times as needed. Let’s let x = a number. Then 1 x =^ its^ reciprocal.
388 CHAPTER 6 Rational Expressions
T T T T T T T ______ , 9 #^ ______ = 1
T T T T T T T ______ + 5 #^ ______ = 6 Finish with:
3. SOLVE and 4. INTERPRET 20. The quotient of a number and 9 times its reciprocal is 1. Find the number(s). Start the solution: 1. UNDERSTAND the problem. Reread it as many times as needed. Let’s let x = a number. Then 1 x = its reciprocal. 2. TRANSLATE into an equation. (Fill in the blanks below.)
Solve. See Example 4.
27. An experienced roofer can roof a house in 26 hours. A beginning roofer needs 39 hours to complete the same job. Find how long it takes for the two to do the job together. 28. Alan Cantrell can word process a research paper in 6 hours. With Steve Isaac’s help, the paper can be processed in 4 hours. Find how long it takes Steve to word process the paper alone. 29. Three postal workers can sort a stack of mail in 20 minutes, 30 minutes, and 60 minutes, respectively. Find how long it takes them to sort the mail if all three work together. 30. A new printing press can print newspapers twice as fast as the old one can. The old one can print the afternoon edition in 4 hours. Find how long it takes to print the afternoon edition if both printers are operating. Solve. See Example 5. 31. Mattie Evans drove 150 miles in the same amount of time that it took a turbopropeller plane to travel 600 miles. The speed of the plane was 150 mph faster than the speed of the car. Find the speed of the plane. 32. An F-100 plane and a Toyota truck leave the same town at sun- rise and head for a town 450 miles away. The speed of the plane is three times the speed of the truck, and the plane arrives 6 hours ahead of the truck. Find the speed of the truck. 33. The speed of Lazy River’s current is 5 mph. If a boat travels 20 miles downstream in the same time that it takes to travel 10 miles upstream, find the speed of the boat in still water. 34. The speed of a boat in still water is 24 mph. If the boat travels 54 miles upstream in the same time that it takes to travel 90 miles downstream, find the speed of the current.
MIXED PRACTICE Solve.
35. The sum of the reciprocals of two consecutive integers is - 1556. Find the two integers. 36. The sum of the reciprocals of two consecutive odd integers is 20 99 . Find the two integers. 37. One hose can fill a goldfish pond in 45 minutes, and two hoses can fill the same pond in 20 minutes. Find how long it takes the second hose alone to fill the pond. 38. If Sarah Clark can do a job in 5 hours and Dick Belli and Sarah working together can do the same job in 2 hours, find how long it takes Dick to do the job alone. 39. Two trains going in opposite directions leave at the same time. One train travels 15 mph faster than the other. In 6 hours, the trains are 630 miles apart. Find the speed of each. 2. TRANSLATE into an equation. (Fill in the blanks below.) a number plus^5 times^
its reciprocal is^6
a number
divided by 9 times^
its reciprocal is^1
Finish with:
3. SOLVE and 4. INTERPRET 21. If a number is added to the numerator of^1241 and twice the
number is added to the denominator of^1241 , the resulting fraction is equivalent to^1 3
. Find the number. 22. If a number is subtracted from the numerator of^138 and added to the denominator of^13 8 , the resulting fraction is equivalent to 2
- Find the number. Solve. See Example 3. 23. An Arabian camel can drink 15 gallons of water in 10 min- utes. At this rate, how much water can the camel drink in 3 minutes? ( Source: Grolier, Inc.) 24. An Arabian camel can travel 20 miles in 8 hours, carrying a 300-pound load on its back. At this rate, how far can the camel travel in 10 hours? ( Source: Grolier, Inc.) 25. In 2010, 13.1 out of every 100 Coast Guard personnel were women. If there were 42,358 total Coast Guard personnel on active duty, estimate the number of women. Round to the nearest whole. ( Source: Women in Military Service for America Memorial Foundation, Inc.) 26. In 2010, 185 out of every 200 marine personnel were men. If there were 202,441 total marine personnel in 2010, estimate the number of men. Round to the nearest whole. ( Source: Women in Military Service for America Memorial Founda- tion, Inc.)
390 CHAPTER 6 Rational Expressions
66. A doctor recorded a body-mass index of 47 on a patient’s chart. Later, a nurse notices that the doctor recorded the patient’s weight as 240 pounds but neglected to record the patient’s height. Explain how the nurse can use the informa- tion from the chart to find the patient’s height. Then find the height. In physics, when the source of a sound is traveling toward an ob- server, the relationship between the actual pitch a of the sound and the pitch h that the observer hears due to the Doppler effect is de- scribed by the formula h = a 1 - s 770
, where s is the speed of the
sound source in miles per hour. Use this formula to answer Exer- cise 67 and 68.
67. An emergency vehicle has a single-tone siren with the pitch of the musical note E. As it approaches an observer standing by the road, the vehicle is traveling 50 mph. Is the pitch that the observer hears due to the Doppler effect lower or higher than the actual pitch? To which musical note is the pitch that the observer hears closest? Pitch of an Octave of Musical Notes in Hertz (Hz) Note Pitch Middle C 261. D 293. E 329. F 349. G 392. A 440. B 493. Note: Greater numbers indicate higher pitches (acoustically). ( Source: American Standards Association) 68. Suppose an emergency van has a single-tone siren with the pitch of the musical note G. If the van is traveling at 80 mph approaching a standing observer, name the pitch the observer hears (rounded to the nearest tenth) and the musical note closest to that pitch. In electronics, the relationship among the resistances R 1 and R 2 of two resistors wired in a parallel circuit and their combined resis- tance R is described by the formula (^) R^1 = (^) R^1 1
. Use this formula to solve Exercises 69 through 71. 69. If the combined resistance is 2 ohms and one of the two resistances is 3 ohms, find the other resistance. 70. Find the combined resistance of two resistors of 12 ohms each when they are wired in a parallel circuit. 71. The relationship among resistance of two resistors wired in a parallel circuit and their combined resistance may be ex- tended to three resistors of resistances R 1 , R 2 , and R 3. Write an equation you believe may describe the relationship and use it to find the combined resistance if R 1 is 5, R 2 is 6, and R 3 is 2. 72. For the formula^1 x = (^1) y + (^1) z - (^) w^1 , find x if y = 2, z = 7, and w = 6.
OBJECTIVE
1 Solving Problems Involving Direct Variation
A very familiar example of direct variation is the relationship of the circumference C of a circle to its radius r. The formula C = 2 p r expresses that the circumference is always 2 p times the radius. In other words, C is always a constant multiple 12 p 2 of r. Because it is, we say that C varies directly as r , that C varies directly with r , or that C is directly proportional to r****.
C 2 p r constant
OBJECTIVES 1 Solve Problems Involving Direct Variation. 2 Solve Problems Involving Inverse Variation. 3 Solve Problems Involving Joint Variation. 4 Solve Problems Involving Combined Variation.
6.7 Variation and Problem Solving
