Real-World Problems:
Rates and Unit Rates
5.2
Lesson Objective
Vocabulary
Le
• Solve problems involving rates and unit rates.
average speed
arn Solve simple word problems involving rates and unit rates.
A machine can pack 70 boxes of spaghetti in 5 minutes.
At this rate, how many boxes of spaghetti can it pack in 8 minutes?
The machine can pack the same number of boxes every minute.
70 boxes
5 min
1 min
?
5 min
70 boxes
1 min
70
5 14 boxes
5
The unit rate is 14 boxes
per minute.
The machine can pack 14 boxes of spaghetti in 1 minute.
8 min
?
8 min
8 3 14 5 112 boxes
14 boxes
112 boxes
5
1 min
8 min
The machine can pack 112 boxes of spaghetti in 8 minutes.
168
Chapter 5 Rates
Guided Practice
Solve.
1 A unicycle wheel makes 196 revolutions in 7 minutes.
a) At this rate, how many revolutions does it make in 1 minute?
The unicycle wheel makes the same number of revolutions every minute.
196 revolutions
7 min
1 min
?
revolutions
?
7 min
?
5
?
1 min
It makes
revolutions
?
?
The unit rate is
?
revolutions
per minute.
revolutions
revolutions in 1 minute.
b) At this rate, how many revolutions does the unicycle wheel make in 15 minutes?
15 min
?
?
?
min
?
3
The unicycle makes
?
5
?
revolutions
revolutions
revolutions in 15 minutes.
2 Megan babysits for 5 hours and earns $60.
a) At this rate, how much does she earn in 1 hour?
?
hours
$ ?
1 hour
$ ?
She earns $
?
4
?
5 $ ?
in 1 hour.
b) At this rate, how much will Megan earn if she babysits for 14 hours?
14 hours
?
3 $ ?
Megan will earn $ ?
5 $ ?
if she babysits for 14 hours.
Lesson 5.2 Real-World Problems: Rates and Unit Rates
169
Le
arn Read a table to find the information to solve multi-step rate problems.
The table shows the fees at a parking lot.
First Hour
Free
Second Hour
$1.75
After the Second Hour
$2.50 per hour
Ben parked his car there from 9 A.M. to 2 p.M. on the same day.
How much did he pay for parking?
Total number of hours 5 5 h
Parking fee for first hour 5 $0
Parking fee for second hour 5 $1.75
Parking fee for last three hours 5 3 3 $2.50
5 $7.50
Total parking fee5 $0 1 $1.75 1 $7.50
5 $9.25
Ben paid $9.25 for parking.
Guided Practice
Solve.
3 The table shows the charges for renting a bicycle.
Tom rented a bicycle from 10 A.M. to 2 P.M. on the same day.
How much did he pay for renting the bicycle?
Total number of hours 5
Charge for first hour 5 $ ?
Charge for each additional 1 hour5 2 3 Cost for each additional 2 hour
?
h
For Every Additional
1
5 2 3 $ ?
5$
Charge for last three hours5
170
?
1 $ ?
5 $ ?
Tom paid $ ? for renting the bicycle.
Chapter 5 Rates
?
3 $ ?
5 $ ?
Total charge 5 $ ?
First Hour
$3.00
1
Hour
2
$2.50
Le
arn Solve multi-step word problems involving comparison of unit rates.
Andy needs new batteries for his video game controller. He is trying to decide
between two brands.
A package of two Brand A batteries costs $3.20. The manufacturer claims the
batteries will last for 20 hours.
A package of two Brand B batteries cost $2.80. The manufacturer claims the
batteries will last for 14 hours.
Which battery should Andy buy? Explain why you think so.
First, find the number of hours of battery time Andy will get per dollar.
Brand A:
$3.20
20 h
20
5 6.25 h
3.20
$1
Andy will get 6.25 hours of battery time per dollar.
Brand B:
$2.80
14 h
14
55h
2.80
$1
Andy will get 5 hours of battery time per dollar.
With Brand A, Andy gets 6.25 hours of battery time per dollar. With Brand B, he gets
5 hours of battery time per dollar. Because 6.25 hours  5 hours, Brand A is a better
buy, and he should buy Brand A.
Guided Practice
Solve.
4 Chloe scored 87 points in 5 basketball games, and Fiona scored 45 points in
2 basketball games. Which of the two players scored more points per game?
Explain.
Chloe:
Fiona:
5 games
?
points
1 games
?
455
Chloe scored
?
points per game.
Comparing the number of points each player scored per game,
higher number of points per game.
So,
?
?
points
2 games
?
points
1 games
?
4
?
Fiona scored
?
?
5
?
points
points per game.
scored a
is a better player.
Lesson 5.2 Real-World Problems: Rates and Unit Rates
171
Le
arn Find the distance given the speed and time.
Mr. Anthony drives his truck at a speed of 45 kilometers per hour .
a) At this speed, how far does he travel in 2 hours?
In 1 hour, Mr. Anthony travels 45 kilometers.
In 2 hours, Mr. Anthony travels 45 3 2 5 90 kilometers.
b) At this speed, how far does he travel in 5 hours?
In 5 hours, Mr. Anthony travels 45 3 5 5 225 kilometers.
Speed
Time
Distance
Distance 5 Speed 3 Time
D
S
Math Note
T
D5S3T
The formula relating distance,
speed, and time is often written as:
Distance 5 Rate 3 Time
172
Chapter 5 Rates
A racing car can travel at a speed of 175 kilometers per hour. How far can the racing
car travel in 3 hours?
175 km
Method 1
1h
175 km
3 3 175 5 525 km
1h
3h
3h
The racing car can travel 525 kilometers in 3 hours.
Method 2
?
Speed 5 175 km/h
Distance 5 Speed 3 Time
5 175 3 3
5 525 km
Time 5 3 h
The racing car can travel 525 kilometers in 3 hours.
Guided Practice
Solve.
5 A high-speed train can travel at a speed of 65 meters per second. How far
can the train travel in 2 seconds?
Method 1
1s
?
?
65 m
m
?
1s
3
?
5
The train can travel
?
meters in 2 seconds.
s
?
m
2s
? m
Method 2
Distance 5 Speed 3 Time
5
?
3
5
?
m
Time 5
The train can travel
?
?
Speed 5
?
?
m/s
s
meters in 2 seconds.
Lesson 5.2 Real-World Problems: Rates and Unit Rates
173
Le
arn Find the time given the distance and speed.
Lucas ran round a field at a speed of 8 meters per second. How long did he take to
run a distance of 96 meters?
Method 1
1s
8m
96
= 12 s
8
96 m
Lucas took 12 seconds to run 96 meters.
Method 2
D
Time= Distance ÷ Speed
= 96 ÷ 8
= 12 s
S
T
T=D4S
Lucas took 12 seconds to run 96 meters.
Guided Practice
Solve.
6 The distance between City X and City Y is 216 kilometers. Mr. Thomas rides
his motorcycle at a speed of 54 kilometers per hour. How long does he take
to travel from City X to City Y?
Method 1
?
km
?
km
?
h
?
5
?
Mr. Thomas takes
Method 2
?
?
h
hours to travel from City X to City Y.
Time
5 Distance ÷ Speed
5
?
4
5
?
h
174
Mr. Thomas takes
Chapter 5 Rates
Speed 5
?
?
?
Distance 5
hours to travel from City X to City Y.
?
km
km/h
Le
arn
Find average speed to solve real-world problems.
The speed that an object is traveling at over a distance may not be the same all
the time. For example, a bicycle might speed up or slow down as it travels between
two points over a given amount of time. You can use an average speed to describe
how fast the bicycle is traveling.
Look at the example.
Post A and Post B are 9 meters apart. Post B and Post C are 21 meters apart.
A bicycle travels from Post A to Post B in 2 seconds. Then it travels from Post B
to Post C in 3 seconds. Find the average speed of the bicycle for the distance from
Post A to Post C.
9 m, 2 s
A
21 m, 3 s
B
C
? m/s
The speed between Post A
Average speed is
and Post B is different from
the average distance
the speed between Post B
traveled per unit time.
and Post C.
Average speed 5
Total distance traveled
Total time
Total distance from Post A to Post C5 9 1 21
5 30 m
Total time to travel from Post A to Post C5 2 1 3
55s
Average speed5
Total distance traveled
Total time
5
30
5
5 6 m/s
The average speed of the bicycle is 6 meters per second.
Lesson 5.2 Real-World Problems: Rates and Unit Rates
175
Guided Practice
Solve.
7 The distance between Town P and Town Q is 80 miles, and the distance
1
hours to travel from
2
between Town Q and Town R is 320 miles. A van takes 2
Town P to Town Q. It takes another 5 hours to travel from Town Q to Town R.
Find the average speed of the van for the whole journey.
?
P
mi,
?
?
h
R
?
mi/h
Total distance from Town P to Town R5 80 1
h
Q
?
mi,
5
?
?
mi
Total time taken to travel from Town P to Town R5 2
5
Average speed5
?
?
h
Total distance traveled
Total time
?
?
5
5
?
4
5
?
mi/h or
1
1
2
?
?
mph
?
The average speed of the van is
miles per hour.
8 Celia ran around a 400-meter track two times. It took her 4 minutes to run
around the track once, and 6 minutes to run around it again. Find Celia’s
average speed.
Total distance Celia ran5 2 3
5
?
?
m
Total time taken to run around the track twice 5
5
Average speed5
5
5
176
1
?
min
Total distance traveled
Total time
?
?
?
m/min
Celia’s average speed was
Chapter 5 Rates
?
?
meters per minute.
?