2.6
Applications of Multiplication
of Fractions
2.6
OBJECTIVES
1. Solve applications involving multiplication
of fractions
2. Use appropriate units analysis when solving
applications
Units Analysis
When dividing two denominate numbers, the units are also divided. This yields
a unit in fraction form.
Examples
250 mi 10 gal 360 ft 30 s 250 mi 25 mi
25 mi/gal (read “miles per gallon”)
10 gal
1 gal
360 ft
12 ft/s (”feet per second”)
30 s
When we multiply denominate numbers that have these units in fraction form,
they behave just like fractions.
Examples
25 mi/gal 12 gal 25 mi 12 gal
300 mi
1 gal
1
(If we look at the units, we see that the gallons essentially “cancel” when one is
in the numerator and the other in the denominator)
12 ft/s 60 s/min 12 ft
60 s
720 ft
720 ft/min
1s
1 min
1 min
(again, the seconds cancel, leaving feet in the numerator and minutes in the
denominator)
© 2001 McGraw-Hill Companies
Let’s look at some applications of our work with the multiplication of fractions. In
solving these word problems, we will use the same approach we used earlier with whole
numbers. Let’s review the four-step process introduced in Section 1.2.
Step by Step: Solving Applications Involving
the Multiplication of Fractions
Step 1 Read the problem carefully to determine the given information and
what you are asked to find.
Step 2 Decide upon the operation or operations to be used.
Step 3 Write down the complete statement necessary to solve the problem
and do the calculations.
Step 4 Check to make sure that you have answered the question of the
problem and that your answer seems reasonable.
179
180
CHAPTER 2
MULTIPLYING AND DIVIDING FRACTIONS
Let’s work through some examples, using these steps.
Example 1
An Application Involving Multiplication
1
h
Lisa worked 10 hours per day
for 5 days. How many hours did she work?
4
day
Step 1 We are looking for the total hours Lisa worked.
Step 2 We will multiply the hours per day by the days.
Step 3 10
1 h
41 h
205
1
5 days 5 days h 51 h
4 day
4 day
4
4
Step 4 Note the days cancel, leaving only the unit hours. The units should always be
compared to the desired units from step 1. The answer also seems reasonable. An
answer like 5 hours or 500 hours would not seem reasonable.
CHECK YOURSELF 1
Carlos gets 30 mi/gal in his Miata. How far should he be able to drive with an 11-gal
tank of gas?
In the next example, we will follow the four steps for solving applications, but we won’t
label the steps. You should still think about these steps as we solve the problem.
Example 2
An Application Involving the Multiplication of Mixed Numbers
3
2
A sheet of notepaper is 6 inches (in.) wide by 8 in. long. Find the area of the paper.
4
3
Multiply the given length by the width. This will give the desired area. First, we will
estimate the area
7 in. 9 in. 63 in.2
Now, we will find the exact area.
2
3
26
27
8 in. 6 in. in. in.
3
4
3
4
rectangle is the product of its
length and its width.
117 2
in.
2
1
58 in.2
2
The units (square inches) are units of area. Note from our estimate that the result is
reasonable.
CHECK YOURSELF 2
1
1
A window is 4 feet (ft) high by 2 ft wide. What is its area?
2
3
© 2001 McGraw-Hill Companies
NOTE Recall that the area of a
APPLICATIONS OF MULTIPLICATION OF FRACTIONS
SECTION 2.6
181
The next example reminds us that an abstract number multiplied by a denominate
number will yield the units of the denominate number.
Example 3
NOTE The word “of” indicates
multiplication.
An Application Involving the Multiplication of a Mixed Number
and a Fraction
2
3
A state park contains 38 acres. According to the plan for the park, of the park is to be
3
4
left as a wildlife preserve. How many acres will this be?
3
2
We want to find of 38 acres. We then multiply as shown:
4
3
1
29
2
3
116
3
38 acres 29 acres
4
3
4
3
1
1
CHECK YOURSELF 3
3
A backyard has 25 square yards (yd2) of open space. If Patrick wants to build a
4
2
vegetable garden covering of the open space, how many square yards will this be?
3
We have mentioned the word “of ” indicates multiplication. You should also note that it
indicates that the fraction preceding it is an abstract number (it has no units attached).
There are even occasions, as in the next example, when we are looking at the product of two
abstract numbers.
Example 4
An Application Involving the Multiplication of Fractions
3
2
of the customers will buy meat. Of these, will buy
3
4
at least one package of beef. What portion of the store’s customers will buy beef?
A grocery store survey shows that
© 2001 McGraw-Hill Companies
2
3
Step 1 We know that of the customers will buy meat and that of these customers
3
4
will buy beef.
NOTE Remember, in this
problem, “of” means to
multiply.
Step 2 We wish to know
3 2
of . The operation here is multiplication.
4 3
Step 3 Multiplying, we have
1
1
3
2
1
4
3
2
2
1
Step 4 From step 3 we have the result:
1
of the store’s customers will buy beef.
2
182
CHAPTER 2
MULTIPLYING AND DIVIDING FRACTIONS
CHECK YOURSELF 4
A supermarket survey shows that
2
of the customers will buy lunch meat. Of these,
5
3
will buy boiled ham. What portion of the store’s customers will buy boiled ham?
4
Example 5
An Application Involving the Multiplication of Mixed Numbers
1
Shirley drives at an average speed of 52 miles per hour (mi/h) for 3 h. How far has she
4
1
traveled at the end of 3 h?
4
52
Speed
Time
13
52 13
mi
14
1
169 mi
CHECK YOURSELF 5
A. The scale on a map is 1 inch (in.) 60 miles (mi). What is the distance in miles
1
between two towns that are 3 in. apart on the map?
2
B. Maria is ordering concrete for a new sidewalk that is to be
1
1
yd thick, 22 yd
9
2
1
long, and 1 yd wide. How much concrete should she order if she must order a
3
whole number of cubic yards?
CHECK YOURSELF ANSWERS
1. 330 mi
5. a. 210 mi
1
2. 10 ft2
2
1
3
3. 17 yd2
4.
6
10
1
b. The answer, 3 yd3, is rounded up to 4 yd3.
3
© 2001 McGraw-Hill Companies
NOTE Remember, distance is
the product of speed and time.
mi
1
52 mi
13
3 h
h
h
4
1 h
4
Name
2.6
Exercises
Section
Date
Evaluate the following. Be sure to use the proper units.
1. 36 mi/h 4 h
2. 80 cal/g 5 g
3. 55 joules/s 11 s
4. 5 lb/ft 3 ft
ANSWERS
1.
2.
5. 88 ft/s 1 mi/5280 ft 3600 s/h
6. 24 h/day 3600 s/h 365 days/yr
3.
4.
5.
Solve the following applications.
7. Maria earns $7 per hour. Last week, she worked 9 hours per day for 6 days. What was
her gross pay?
7.
8. The gas tank in Luigi’s Toyota Camry holds 17 gal when full. The car gets 21 migal.
How far can he travel on three full tanks?
9. Recipes. A recipe calls for
6.
2
cup of sugar for each serving. How much sugar is
3
8.
9.
10.
needed for six servings?
11.
3
cup of batter for each serving. If five
4
people are expected for breakfast, how much batter is needed?
10. Recipes. Mom-Mom’s French toast requires
12.
5
6
3
decides to cover only of the dirt, how much sod does he need?
4
11. Gardening. A patch of dirt needs 3 square feet (ft2) of sod to cover it. If Nick
5
6
1
Sheila wants to enlarge her driveway to 2 times its current size, how much concrete
2
will she need?
© 2001 McGraw-Hill Companies
12. Construction. A driveway requires 4 cubic yards (yd3) of concrete to cover it. If
183
ANSWERS
13.
13. Map scales. The scale on a map is 1 inch (in.) 200 miles (mi). What actual
14.
distance, in miles, does
3
in. represent?
8
15.
16.
17.
18.
19.
14. Salary. You make $90 a day on a job. What will you receive for working
3
of a day?
4
15. Size. A lumberyard has a stack of 80 sheets of plywood. If each sheet is
3
in. thick,
4
how high will the stack be?
2
of its monthly income for housing and utilities on
5
average. If the family’s monthly income is $1750, what is spent for housing and
utilities? What amount remains?
16. Family budget. A family uses
3
were registered. Of those registered,
4
5
actually voted. What fraction of those people who were eligible voted?
9
7
of the people in a city own pets. Of those who
10
2
own pets, have dogs. What fraction of those surveyed own dogs?
3
18. Surveys. A survey has found that
1
3
3
4
(yd2) of linoleum must be bought to cover the floor?
19. Area. A kitchen has dimensions 3 yards (yd) by 3 yd. How many square yards
184
© 2001 McGraw-Hill Companies
17. Elections. Of the eligible voters in an election,
ANSWERS
3
4
20.
20. Distance. If you drive at an average speed of 52 miles per hour (mi/h) for 1 h, how
far will you travel?
21.
22.
2
3
21. Distance. A jet flew at an average speed of 540 mi/h on a 4 -h flight. What was the
distance flown?
23.
24.
25.
26.
2
3
2
estimated that of the area will be used for roads. What amount remains to be used
7
for lots?
22. Area. A piece of land that has 11 acres is being subdivided for home lots. It is
23. Circumference. To find the approximate circumference or distance around a circle,
22
we multiply its diameter by . What is the circumference of a circle with a diameter
7
of 21 in.?
24. Area. The length of a rectangle is
square yards?
6
21
yd, and its width is
yd. What is its area in
7
26
1
4
7
8
5
6
25. Volume. Find the volume of a box that measures 2 in. by 3 in. by 4 in.
26. Topsoil. Nico wishes to purchase mulch to cover his garden. The garden measures
© 2001 McGraw-Hill Companies
7
1
1
7 feet (ft) by 10 ft. He wants the mulch to be ft deep. How much mulch should
8
8
3
Nico order if he must order a whole number of cubic feet?
10 18 ft
7 78 ft
185
ANSWERS
27.
The formula for the area of a triangle is
28.
A
h
1
hb
2
b
29.
in which h is the height of the triangle and b is the base.
30.
2
5
1
3
7
8
2
5
27. Find the area of a triangle with a height of 2 in. and a base of 3 in.
28. Find the area of a triangle with a height of 1 in. and a base of 2 in.
29. A recipe calls for the following ingredients:
7
3
2
5
cup of flour, cup of sugar, cup of milk, and teaspoon of salt. This recipe
8
4
3
6
makes eight servings. What amount of each quantity would you use if you wanted to
serve two people?
30. Obtain a map of your state and, using the legend provided, determine the distance
between your state capital and any other city. Would this be the actual distance you
would travel by car if you made the journey? Why or why not?
Answers
1. 144 mi
7 2
11. 2 ft
8
13. 75 mi
23. 66 in.
5. 60 mi/h
15. 60 in.
25. 42
7. $378
5
17.
12
9
in.3
64
9. 4 cups
1
2
2
19. 12 yd
27. 4 in.2
29.
© 2001 McGraw-Hill Companies
21. 2520 mi
3. 605 joules
186