5.6
Applications of Algebraic
Fractions
5.6
OBJECTIVES
1. Solve a word problem that leads to a fractional
equation
2. Apply proportions to the solution of a word problem
Many word problems will lead to fractional equations that must be solved by using the
methods of the previous section. The five steps in solving word problems are, of course, the
same as you saw earlier.
Example 1
Solving a Numerical Application
If one-third of a number is added to three-fourths of that same number, the sum is 26. Find
the number.
Step 1 Read the problem carefully. You want to find the unknown number.
Step 2 Choose a letter to represent the unknown. Let x be the unknown number.
Step 3 Form an equation.
NOTE The equation expresses
the relationship between the
two parts of the number.
1
3
x x 26
3
4
One-third
of number
Three-fourths
of number
Step 4 Solve the equation. Multiply each side (every term) of the equation by 12,
the LCD.
1
3
12 x 12 x 12 26
3
4
Simplifying yields
4x 9x 312
13x 312
x 24
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NOTE Be sure to answer the
The number is 24.
question raised in the problem.
Step 5 Check your solution by returning to the original problem. If the number is 24,
we have
3
1
24 24 8 18 26
3
4
and the solution is verified.
447
448
CHAPTER 5
ALGEBRAIC FRACTIONS
CHECK YOURSELF 1
The sum of two-fifths of a number and one-half of that number is 18. Find the
number.
Number problems that involve reciprocals can be solved by using fractional equations.
Example 2 illustrates this approach.
Example 2
Solving a Numerical Application
3
One number is twice another number. If the sum of their reciprocals is , what are the two
10
numbers?
Step 1 You want to find the two numbers.
Step 2 Let x be one number. Then 2x is the other number.
Twice the first
Step 3
NOTE The reciprocal of a
fraction is the fraction obtained
by switching the numerator and
denominator.
1
x
1
2x
The reciprocal
of the first
number, x
3
10
The reciprocal
of the second
number, 2x
Step 4 The LCD of the fractions is 10x, and so we multiply by 10x.
10x
x 10x2x 10x10
1
1
3
Simplifying, we have
10 5 3x
15 3x
5x
The numbers are 5 and 10.
Step 5
Again check the result by returning to the original problem. If the numbers are 5 and 10, we
have
1
1
21
3
5
10
10
10
The sum of the reciprocals is
3
.
10
CHECK YOURSELF 2
2
One number is 3 times another. If the sum of their reciprocals is , find the two num9
bers.
© 2001 McGraw-Hill Companies
NOTE x was one number, and
2x was the other.
APPLICATIONS OF ALGEBRAIC FRACTIONS
SECTION 5.6
449
The solution of many motion problems will also involve fractional equations. Remember that the key equation for solving all motion problems relates the distance traveled, the
speed or rate, and the time:
Definitions: Motion Problem Relationships
drt
Often we will use this equation in different forms by solving for r or for t. So
r
d
t
or
t
d
r
Example 3
Solving an Application Involving r d
t
Vince took 2 hours (h) longer to drive 225 miles (mi) than he did on a trip of 135 mi. If his
speed was the same both times, how long did each trip take?
225 miles
135 miles
Step 1 You want to find the times taken for the 225-mi trip and for the 135-mi trip.
© 2001 McGraw-Hill Companies
choose your variable to
“suggest” the unknown
quantity—here t for time.
Step 2 Let t be the time for the 135-mi trip (in hours).
2 h longer
NOTE It is often helpful to
Then t 2 is the time for the 225-mi trip.
It is often helpful to arrange the information in tabular form such as that shown.
Distance
Time
NOTE Remember that rate is
135-mi trip
135
t
distance divided by time. The
rightmost column is formed by
using that relationship.
225-mi trip
225
t2
Rate
135
t
225
t2
Step 3 In forming the equation, remember that the speed (or rate) for each trip was the
same. That is the key idea. We can equate the rates for the two trips that were found in step 2.
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CHAPTER 5
ALGEBRAIC FRACTIONS
The two rates are shown in the rightmost column of the table. Thus we can write
135
225
t
t2
NOTE Notice that the equation
is in proportion form. So we
could solve by setting the
product of the means equal to
the product of the extremes.
Step 4 To solve the above equation, multiply each side by t(t 2), the LCD of the
fractions.
t (t 2)
t t(t 2)t 2
135
225
Simplifying, we have
135(t 2) 225t
135t 270 225t
270 90t
t 3h
The time for the 135-mi trip was 3 h, and the time for the 225-mi trip was 5 h. We’ll leave
the check to you.
CHECK YOURSELF 3
Cynthia took 2 h longer to bicycle 75 mi than she did on a trip of 45 mi. If her speed
was the same each time, find the time for each trip.
Example 4 uses the d r t relationship to find the speed.
Example 4
Solving an Application Involving d r t
A train makes a trip of 300 mi in the same time that a bus can travel 250 mi. If the speed of
the train is 10 mi/h faster than the speed of the bus, find the speed of each.
Step 1 You want to find the speeds of the train and of the bus.
Step 2 Let r be the speed (or rate) of the bus (in miles per hour).
Then r 10 is the rate of the train.
Again let’s form a table of the information.
Distance
Rate
Time
300
r 10
250
r
NOTE Remember that time is
Train
300
r 10
distance divided by rate. Here
the rightmost column is found
by using that relationship.
Bus
250
r
© 2001 McGraw-Hill Companies
10 mi/h faster
APPLICATIONS OF ALGEBRAIC FRACTIONS
SECTION 5.6
451
Step 3 To form an equation, remember that the times for the train and bus are the same.
We can equate the expressions for time found in step 2. Again, working from the rightmost
column, we have
250
300
r
r 10
Step 4 We multiply each term by r(r 10), the LCD of the fractions.
1
r(r 10)
1
250
300
r(r 10)
r
r 10
1
1
Simplifying, we have
250(r 10) 300r
250r 2500 300r
2500 50r
r 50 mi/h
NOTE Remember to find the
rates of both vehicles.
The rate of the bus is 50 mi/h, and the rate of the train is 60 mi/h. You can check this result.
CHECK YOURSELF 4
A car makes a trip of 280 mi in the same time that a truck travels 245 mi. If the speed
of the truck is 5 mi/h slower than that of the car, find the speed of each.
The next example involves fractions in decimal form. Mixture problems often use
percentages, and those percentages can be written as decimals. Example 5 illustrates this
method.
Example 5
Solving an Application Involving Solutions
A solution of antifreeze is 20% alcohol. How much pure alcohol must be added to 12 quarts
(qt) of the solution to make a 40% solution?
Step 1 You want to find the number of quarts of pure alcohol that must be added.
© 2001 McGraw-Hill Companies
Step 2 Let x be the number of quarts of pure alcohol to be added.
Step 3 To form our equation, note that the amount of alcohol present before mixing must
be the same as the amount in the combined solution.
A picture will help.
12 qt
20%
x qt
100%
12 x qt
40%
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CHAPTER 5
ALGEBRAIC FRACTIONS
So
The amount of
alcohol in the
first solution
(20% is 0.20)
as decimals in the equation.
12(0.20) x(1.00) (12 x)(0.40)
NOTE Express the percentages
The amount of
pure alcohol
(“pure” is 100%,
or 1.00)
The amount of
alcohol in the
mixture
Step 4 Most students prefer to clear the decimals at this stage. It’s easy here—
multiplying by 100 will move the decimal point two places to the right. We then have
12(20) x(100) (12 x)(40)
240 100x 480 40x
60x 240
x 4 qt
CHECK YOURSELF 5
How much pure alcohol must be added to 500 cubic centimeters (cm3) of a 40%
alcohol mixture to make a solution that is 80% alcohol?
There are many types of applications that lead to proportions in their solution. Typically
these applications will involve a common ratio, such as miles to gallons or miles to hours,
and they can be solved with three basic steps.
Step by Step:
Step 1
Step 2
Step 3
To Solve an Application by Using Proportions
Assign a variable to represent the unknown quantity.
Write a proportion, using the known and unknown quantities. Be sure
each ratio involves the same units.
Solve the proportion written in step 2 for the unknown quantity.
Example 6 illustrates this approach.
Example 6
Solving an Application Using Proportions
Step 1 Assign a variable to represent the unknown quantity. Let x be the number of
gallons of gas that will be used on the 385-mi trip.
Step 2 Write a proportion. Note that the ratio of miles to gallons must stay the same.
Miles
Miles
385
105
3
x
Gallons
Gallons
© 2001 McGraw-Hill Companies
A car uses 3 gallons (gal) of gas to travel 105 miles (mi). At that mileage rate, how many
gallons will be used on a trip of 385 mi?
APPLICATIONS OF ALGEBRAIC FRACTIONS
SECTION 5.6
453
Step 3 Solve the proportion. The product of the extremes is equal to the product of the
means.
105x 3 385
105x 1155
1155
105x
105
105
NOTE To verify your solution,
return to the original problem
and check that the two ratios
are equivalent.
x 11 gal
So 11 gal of gas will be used for the 385-mi trip.
CHECK YOURSELF 6
A car uses 8 liters (L) of gasoline in traveling 100 kilometers (km). At that rate, how
many liters of gas will be used on a trip of 250 km?
Proportions can also be used to solve problems in which a quantity is divided by using
a specific ratio. Example 7 shows how.
Example 7
Solving an Application Using Proportions
A piece of wire 60 inches (in.) long is to be cut into two pieces whose lengths have the ratio
5 to 7. Find the length of each piece.
Step 1 Let x represent the length of the shorter piece. Then 60 x is the length of the
longer piece.
NOTE A picture of the problem
Shorter
Longer
always helps.
60 x
x
60
5
Step 2 The two pieces have the ratio , so
7
NOTE On the left and right,
© 2001 McGraw-Hill Companies
we have the ratio of the length
of the shorter piece to that of
the longer piece.
5
x
60 x
7
Step 3 Solving as before, we get
7x (60 x)5
7x 300 5x
12x 300
x 25
(Shorter piece)
60 x 35
(Longer piece)
The pieces have lengths 25 in. and 35 in.
CHAPTER 5
ALGEBRAIC FRACTIONS
CHECK YOURSELF 7
A board 21 feet (ft) long is to be cut into two pieces so that the ratio of their lengths
is 3 to 4. Find the lengths of the two pieces.
CHECK YOURSELF ANSWERS
1. The number is 20.
2. The numbers are 6 and 18.
3. 75-mi trip: 5 h; 45-mi trip: 3 h
4. Car: 40 mi/h; truck: 35 mi/h
6. 20 L
7. 9 ft; 12 ft
5. 1000 cm3
© 2001 McGraw-Hill Companies
454
Name
5.6
Exercises
Section
Date
Solve the following word problems.
1. Adding numbers. If two-thirds of a number is added to one-half of that number, the
ANSWERS
sum is 35. Find the number.
2. Subtracting numbers. If one-third of a number is subtracted from three-fourths of
that number, the difference is 15. What is the number?
1.
2.
3. Subtracting numbers. If one-fourth of a number is subtracted from two-fifths of a
3.
number, the difference is 3. Find the number.
4.
4. Adding numbers. If five-sixths of a number is added to one-fifth of the number, the
5.
sum is 31. What is the number?
5. Consecutive integers. If one-third of an integer is added to one-half of the next
6.
consecutive integer, the sum is 13. What are the two integers?
7.
6. Consecutive integers. If one-half of one integer is subtracted from three-fifths of the
next consecutive integer, the difference is 3. What are the two integers?
8.
9.
7. Reciprocals. One number is twice another number. If the sum of their reciprocals is
1
, find the two numbers.
4
10.
1
6
8. Reciprocals. One number is 3 times another. If the sum of their reciprocals is , find
the two numbers.
11.
12.
9. Reciprocals. One number is 4 times another. If the sum of their reciprocals is
find the two numbers.
5
,
12
13.
14.
4
10. Reciprocals. One number is 3 times another. If the sum of their reciprocals is ,
15
what are the two numbers?
15.
11. Reciprocals. One number is 5 times another number. If the sum of their reciprocals
is
6
, what are the two numbers?
35
12. Reciprocals. One number is 4 times another. The sum of their reciprocals is
What are the two numbers?
5
.
24
© 2001 McGraw-Hill Companies
13. Reciprocals. If the reciprocal of 5 times a number is subtracted from the reciprocal
of that number, the result is
4
. What is the number?
25
14. Reciprocals. If the reciprocal of a number is added to 4 times the reciprocal of that
5
number, the result is . Find the number.
9
15. Driving rate. Lee can ride his bicycle 50 miles (mi) in the same time it takes him
to drive 125 mi. If his driving rate is 30 mi/h faster than his rate bicycling, find
each rate.
455
ANSWERS
16. Running rate. Tina can run 12 mi in the same time it takes her to bicycle 72 mi. If
16.
her bicycling rate is 20 mi/h faster than her running rate, find each rate.
17.
18.
19.
20.
21.
22.
23.
17. Driving rate. An express bus can travel 275 mi in the same time that it takes a local
bus to travel 225 mi. If the rate of the express bus is 10 mi/h faster than that of the
local bus, find the rate for each bus.
24.
18. Flying time. A light plane took 1 hour (h) longer to travel 450 mi on the first portion
of a trip than it took to fly 300 mi on the second. If the speed was the same for each
portion, what was the flying time for each part of the trip?
19. Train speed. A passenger train can travel 325 mi in the same time a freight train
takes to travel 200 mi. If the speed of the passenger train is 25 mi/h faster than the
speed of the freight, find the speed of each.
20. Flying time. A small business jet took 1 h longer to fly 810 mi on the first part of a
flight than to fly 540 mi on the second portion. If the jet’s rate was the same for each
leg of the flight, what was the flying time for each leg?
21. Driving time. Charles took 2 h longer to drive 240 mi on the first day of a vacation
trip than to drive 144 mi on the second day. If his average driving rate was the same
on both days, what was his driving time for each of the days?
22. Driving time. Ariana took 2 h longer to drive 360 mi on the first day of a trip than
she took to drive 270 mi on the second day. If her speed was the same on both days,
what was the driving time each day?
480 mi. If the plane’s rate was the same on each trip, what was the time of each flight?
24. Traveling time. A train travels 80 mi in the same time that a light plane can travel
280 mi. If the speed of the plane is 100 mi/h faster than that of the train, find each of
the rates.
456
© 2001 McGraw-Hill Companies
23. Flying time. An airplane took 3 h longer to fly 1200 mi than it took for a flight of
ANSWERS
25. Canoeing time. Jan and Tariq took a canoeing trip, traveling 6 mi upstream against a
2 mi/h current. They then returned to the same point downstream. If their entire trip
took 4 h, how fast can they paddle in still water? [Hint: If r is their rate (in miles per
hour) in still water, their rate upstream is r 2 and their rate downstream is r 2.].
26. Flying speed. A plane flies 720 mi against a steady 30 mi/h headwind and then
returns to the same point with the wind. If the entire trip takes 10 h, what is the
plane’s speed in still air?
27. Alcohol solution. How much pure alcohol must be added to 40 ounces (oz) of a
25% solution to produce a mixture that is 40% alcohol?
28. Mixtures. How many centiliters (cL) of pure acid must be added to 200 cL of a 40%
acid solution to produce a 50% solution?
25.
26.
27.
28.
29.
30.
31.
32.
33.
29. Speed conversion. A speed of 60 miles per hour (mi/h) corresponds to 88 feet per
second (ft/s). If a light plane’s speed is 150 mi/h, what is its speed in feet per second?
34.
30. Cost. If 342 cups of coffee can be made from 9 pounds (lb) of coffee, how many
cups can be made from 6 lb of coffee?
35.
31. Fuel consumption. A car uses 5 gallons (gal) of gasoline on a trip of 160 mi. At the
same mileage rate, how much gasoline will a 384-mi trip require?
32. Fuel consumption. A car uses 12 liters (L) of gasoline in traveling 150 kilometers
© 2001 McGraw-Hill Companies
(km). At that rate, how many liters of gasoline will be used in a trip of 400 km?
33. Yearly earnings. Sveta earns $13,500 commission in 20 weeks in her new sales
position. At that rate, how much will she earn in 1 year (52 weeks)?
34. Investment earning. Kevin earned $165 interest for 1 year on an investment of
$1500. At the same rate, what amount of interest would be earned by an investment
of $2500?
35. Insect control. A company is selling a natural insect control that mixes ladybug
beetles and praying mantises in the ratio of 7 to 4. If there are a total of 110 insects
per package, how many of each type of insect is in a package?
457
ANSWERS
36.
36. Individual height. A woman casts a shadow of 4 ft. At the same time, a 72-ft
building casts a shadow of 48 ft. How tall is the woman?
37.
38.
37. Consumer affairs. A brother and sister are to divide an inheritance of $12,000 in
the ratio of 2 to 3. What amount will each receive?
38. Taxes. In Bucks County, the property tax rate is $25.32 per $1000 of assessed value.
Answers
1. 30
3. 20
5. 15, 16
7. 6, 12
9. 3, 12
11. 7, 35
15. 20 mi/h bicycling, 50 mi/h driving
17. Express 55 mi/h, local 45 mi/h
19. Freight 40 mi/h, passenger 65 mi/h
21. 5 h, 3 h
23. 5 h, 2 h
25. 4 mi/h
27. 10 oz
29. 220 ft/s
31. 12 gal
33. $35,100
35. 70 ladybugs, 40 praying mantises
37. Brother $4800, sister $7200
458
13. 5
© 2001 McGraw-Hill Companies
If a house and property have a value of $128,000, find the tax the owner will have to
pay.