Chapter 6 Study Guide
Lessons 6-1 to 6-3
Lesson 6-1: Ratios
Example 1 Write Ratios in Simplest Form
Express the ratio 8 golden retrievers out of 20 dogs as a fraction in simplest form.
Explain its meaning.
4
8
2
=
20
5
Divide the numerator and denominator by the GCF, 4.
4
The ratio of golden retrievers to dogs is 2 to 5. This means that for every 5 dogs, 2 of
2
them are golden retrievers. Also, 5 of the dogs are golden retrievers.
2
The ratio representing 2 out of 5 can be written as: 2 to 5, 2:5, or 5
Real-World Example 2
Write Ratios as Fractions
LUNCH Six out of every 10 students who bought lunch today chose pizza. Express
this ratio as a fraction in simplest form. Explain its meaning.
÷2
6 3
10 = 5
Divide the numerator and
denominator by the GCF, 2.
÷2
The ratio of students who chose pizza is 3 to 5. This means that for every 5 students who
3
bought lunch, 3 chose pizza. Also, 5 of students who bought lunch chose pizza.
3
The ratio representing 3 out of 5 can be written as: 3 to 5, 3:5, or 5
Example 3
Fractions
Write Ratios Comparing Different Units of Measurements as
Express the ratio 2 yards to 4 feet as a fraction in simplest form.
2 yards 6 feet
=
Convert 2 yards to feet.
4feet 4 feet
3 feet
=
Divide the numerator and denominator by the GCF, 2.
2 feet
Written in simplest form, the ratio is 3 to 2.
1
Lesson 6-2: Unit Rates
Example 1 Find Unit Rates
Express each rate as a unit rate. Round to the nearest tenth, if necessary.
a. 10 minutes to run 5 laps
Write the rate that compares the
number of minutes to the number of
laps. Then divide to find the unit rate.
5
10 minutes 2 minutes
5 laps = 1 lap
So, the rate is 2 minutes per lap.
5
b. $3.48 for 6 granola bars
Write the rate that compares the cost
to the number of bars. Then divide to
find the unit rate.
So, the cost is $0.58 per granola bar.
6
$3.48 $0.58
6 bars = 1 bar
6
Real-World Example 2
Compare Unit Rates
SHOPPING A 20-ounce box of cereal costs $3.80, and a 36-ounce box of cereal
costs $6.45. Which box has the lower cost per ounce of cereal?
First, find the unit rates of the two boxes of cereal. Then compare them.
 20
3.80
0.19
dollars
dollars
=
20
ounces1ounce
Divide the numerator and
denominator by 20.
 20
For the 20-ounce box, the unit rate is $0.19 per ounce.
 36
6
.
4
5
d
o
l
l
a
r
s0
.
1
7
9
d
o
l
l
a
r
s

3
6
o
u
n
c
e
s 1
o
u
n
c
e
Divide the numerator and
denominator by 36.
 36
For the 36-ounce box, the unit rate is about $0.179 per ounce.
Since $0.179  $0.19, the 36-ounce box of cereal has a lower cost per ounce.
2
Real-World Example 3
Use Unit Rates
CRAFTS Sam can make 11 bracelets in 5 hours. How many bracelets can she
make in 8 hours?
Find the unit rate. Then multiply this unit rate by 8 to find the number of bracelets she
can make in 8 hours.
11 bracelets in 5 hours =
11 bracelets ÷ 5 2.2 bracelets
5 hours ÷ 5 or 1 hour
2.2 bracelets
1 hour  8 hours = 17.6
Find the unit rate.
Divide out the common units.
Sam can make 17.6 bracelets in 8 hours.
Lesson 6-2 (continued): Complex Fractions and Unit Rates
Example 1
Simplify a Complex Fraction
Simplify
a.)
1
1
1
2
= 1÷
1
2
=
1
2
1 2
∙
1 1
= 2
b.)
1
4
2
Write the complex fraction as a division problem.
Multiply by the reciprocal of
1
2
2
, which is 1
Simplify
1
4
2
=
=
=
1
4
÷2
1 1
∙
4 2
1
8
Write the complex fraction as a division problem.
Multiply by the reciprocal of
2
1
1
, which is 2
Simplify
3
Example 2
Real World Example: Find a Unit Rate
1
1
John can run 1 3 𝑚𝑖𝑙𝑒𝑠 in 4 ℎ𝑜𝑢𝑟. Find his average speed in miles per hour.
1
13 𝑚𝑖
1
ℎ𝑟
4
1
1
3
4
= 1 ÷
∙
16
1
4
4
, which is 1
Simplify
3
=5
1
miles per hour
3
Example 3
Write as a Fraction in Simplest Form
1
33 %
as a fraction in simplest form.
3
1
1
333
3
100
33 % =
1
100
Multiply by the reciprocal of
3 1
=
333
Change the mixed number to an improper fraction.
4 4
=
Write
Write the complex fraction as a division problem.
Definition of percent.
1
= 333 ÷ 100
1
=
100
3
∙
Write the complex fraction as a division problem.
Change the mixed number to an improper fraction.
1
Multiply by the reciprocal of 100, which is
100
1
100
1
=
1
3
Simplify
4
Lesson 6-3: Converting Rates and Measurements
Example 1 Convert Measurements Between Systems
Complete each conversion.
a. 10 feet to meters
Use 1 ft ≈ 0.305m.
10 ft 
10 𝑓𝑡

10 𝑓𝑡
•
1

•
1
0.305 𝑚
1 𝑓𝑡
0.305 𝑚
1 𝑓𝑡
3.05 𝑚
or 3.05 m
1
Multiply by
0.305 𝑚
1 𝑓𝑡
Divide out common units, leaving the desired unit, meter.
Simplify.
So, 10 feet is approximately 3.05 meters.
b. 55 kilograms to pounds
Use 1 kg ≈ 2.203 lb.
55 kg 


55 𝑘𝑔
1
55 𝑘𝑔
•
•
1
2.203 𝑙𝑏
2.203 𝑙𝑏
2.203 𝑙𝑏
.
1 𝑘𝑔
Divide out common units, leaving the desired unit, pound.
1 𝑘𝑔
121.165 𝑙𝑏
1
Multiply by
1 𝑘𝑔
or 121.165 lb Simplify.
So, 55 kilograms is approximately 121.165 pounds.
Real-World Example 2
Use Dimensional Analysis
POOLS A community swimming pool fills at a rate of 70 gallons per minute. How
many gallons per hour is this?
Step 1
You need to convert gallons per minute to gallons per hour. Choose a
conversion factor that converts minutes to hours, with hours in the denominator.
gallons minute gallons
minute  hour = hour
So, use
Step 2
60 min
1h .
Multiply.
70 gal 70 gal 60 min
min = min  1 h
70 gal 60 min
= min  1 h
Multiply by
60 min
1h .
Divide out common units.
= 4,200 gallons per hour
So, the pool will fill at a rate of 4,200 gallons per hour.
5
Real-World Example 3
Convert Rates
TRAVEL A car is traveling at a rate of 60 miles per hour. How many feet per
second is this?
You need to convert miles per hour to feet per second.
Use 5280 ft = 1 mi and 3600 s = 1 h.
60 mi
60 mi 5280ft
1h
=


1h
1h
1mi
3600s
1
=
5280 ft
1h
Multiply by 1 mi and 3600 s.
8
8
6
0m
i 5
2
8
0ft 1h
×
×
1h
1m
i 3
6
0
0s
Divide out the common factors and units.
6
0
1
88 ft
=
s
Simplify.
So, 60 miles per hour is equivalent to 88 feet per second.
Real-World Example 4
Convert Rates Between Systems
AIRPLANES An airplane travels at a speed of 720 kilometers per hour. Determine
how many feet per second the airplane travels.
To convert kilometers to feet, use 1 km ≈ 0.621 mi and 5280 ft = 1 mi.
To convert hours to seconds, use 1 h = 60 min and 1 min = 60 s.
720 km 0.621 mi 5280 ft
1h
1 min




1h
1 km
1 mi 60 min 60 s
=
720 km 0.621 mi 5280 ft
1h
1 min
1 h  1 km  1 mi  60 min  60 s
Divide out common units.
=
2
,3
6
0
,7
9
3
.6ft
3
6
0
0s
Multiply.
=
655.776ft
1s
Divide.
So, the airplane travels about 655.776 feet per second.
6