Ch. 7 Learning Goal: Ratios & Proportions
• Learn to find equivalent ratios to create proportions (7-1)
• Learn to work with rates and ratios (7-2)
• Learn to use one or more conversion factors to solve rate
problems (7-3)
• Learn to solve proportions (7-4)
• Learn to identify and create dilations of plane figures (7-5)
• Learn to determine whether figures are similar, to use scale
factors, and to find missing dimensions similar figures (7-6)
• Learn to make comparisons between and find dimensions of
scale drawings and actual objects (7-7)
• Learn to make comparisons between and find dimensions of
scale models and actual objects (7-8)
• Learn to make scale models of solid figures (7-9)
Page 348 #6-18 Answers
7-3 Analyze Units
Pre-Algebra Homework
Page 353
#8-14
Pre-Algebra
7-3
7-3 Analyze
AnalyzeUnits
Units
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
Pre-Algebra
7-3 Analyze Units
Warm Up
Find each unit rate.
1. Jump rope 192 times in 6 minutes
32 jumps/min
2. Four pounds of bananas for $2.36
$0.59/lb
3. 16 anchor bolts for $18.56
$1.16/bolt
4. 288 movies on 9 shelves
32 movies/shelf
Pre-Algebra
7-3 Analyze Units
Problem of the Day
Replace each • with a digit from 0 to 6 to
make equivalent ratios. Use each digit only
once.
•• = •
•• ••
Pre-Algebra
13 = 4
Possible answer:
65 20
7-3 Analyze Units
Today’s Learning Goal Assignment
Learn to use one
or more conversion
factors to solve
rate problems.
Pre-Algebra
7-3 Analyze Units
Vocabulary
conversion factor
Pre-Algebra
7-3 Analyze Units
You can measure the speed of an object
by using a strobe lamp and a camera in a
dark room. Each time the lamp flashes,
the camera records the object’s position.
Problems often require dimensional
analysis, also called unit analysis, to
convert from one unit to another.
Pre-Algebra
7-3 Analyze Units
To convert units, multiply by one or more
ratios of equal quantities called
conversion factors.
For example, to convert inches to feet
you would use the ratio below as a
conversion factor.
1 ft
12 in
Pre-Algebra
7-3 Analyze Units
Multiplying by a conversion factor is like
multiplying by a fraction that reduces to
1, such as 5 .
5
1 ft = 12 in. , or 1 ft = 1
12 in. 12 in.
1 ft
Pre-Algebra
7-3 Analyze Units
Helpful Hint
The conversion factor
• must introduce the unit desired in the answer
and
• must cancel the original unit so that the unit
desired is all that remains.
Pre-Algebra
7-3 Analyze Units
Additional Example 1: Finding Conversion Factors
Find the appropriate factor for each conversion.
A. feet to yards
There are 3 feet in 1 yard. To convert feet to
yards, multiply the number of feet by 1 yd.
3 ft
B. pounds to ounces
There are 16 ounces in 1 pound. To convert
pounds to ounces, multiply the number of
pounds by 16 oz .
1 lb
Pre-Algebra
7-3 Analyze Units
Try This: Example 1
Find the appropriate factor for each conversion.
A. minutes to seconds
There are 60 seconds in 1 minute. To
convert minutes to seconds, multiply the
number of minutes by 60 sec .
1 min
B. hours to days
There are 24 hours in 1 day. To convert
hours to days, multiply the number hours by
1 day .
24 h
Pre-Algebra
7-3 Analyze Units
Additional Example 2: Using Conversion Factors to
Solve Problems
The average American uses 580 pounds of
paper per year. Find the number of pounds of
paper the average American uses per month,
to the nearest tenth.
The problem gives the ratio 580 pounds to 1 year
and asks for an answer in pounds per month.
580 lb
1 yr
= 580 lb
12 mo
1 yr
12 mo
Multiply the ratio by the
conversion factor
Cancel yr units.
= 48.3 lb per month Divide 580 by 12.
Pre-Algebra
7-3 Analyze Units
Additional Example 2 Continued
The average American uses 580 pounds of
paper per year. Find the number of pounds of
paper the average American uses per month,
to the nearest tenth.
The average American uses 48.3 pounds of paper
per month.
Pre-Algebra
7-3 Analyze Units
Try This: Example 2
Sam drives his car 23,000 miles per year. Find
the number of miles he drives per month.
The problem gives the ratio 23,000 miles to 1 year
and asks for an answer in miles per month.
23,000 mi
1 yr
1 yr
12 mo
= 23,000 mi
12 mo
= 1916.6 per month
Multiply the ratio by the
conversion factor
Cancel yr units.
Divide 23,000 by 12.
Sam drives his car about 1917 miles per month.
Pre-Algebra
7-3 Analyze Units
Additional Example 3: Problem Solving Application
A car traveled 60 miles on a road in 2 hours. How
many feet per second was the car traveling?
1
Understand the Problem
The problem is stated in units of miles and
hours. The question asks for the answer in
units of feet and seconds. You will need to use
several conversion factors.
List the important information:
5280 ft
• Miles to feet
1 mi
1h
• Hours to minutes
60 min
1 min
• Minutes to seconds
60 s
Pre-Algebra
7-3 Analyze Units
Additional Example 3 Continued
2
Make a Plan
Multiply by each conversion factor
separately, or simplify the problem and
multiply by several conversion factors at
once.
Pre-Algebra
7-3 Analyze Units
3
Solve
First, convert 60 miles in 2 hours into a unit
rate.
60 mi = (60÷2) mi = 30 mi
2h
1h
(2÷2) h
Create a single conversion factor to convert
hours directly to seconds:
hours to minutes
1h
minutes to seconds
60 min
1 min
60 s
hours to seconds = 1 h • 1 min = 1 h
3600 s
60 min
60 s
30 mi •
1h
Pre-Algebra
5280 ft • 1 h
1 mi
3600 s
Set up the conversion factors.
7-3 Analyze Units
3 Solve
Do not include the numbers yet.
Notice what happens to the units.
mi • ft • h
mi s
h
Simplify. Only ft remains.
s
30 mi • 5280 ft • 1 h
Multiply.
3600 s
1 mi
1h
30 • 5280 ft • 1 = 158,400 ft = 44 ft
1 • 1 • 3600 s
3600 s
1s
The car was traveling 44 feet per second.
Pre-Algebra
7-3 Analyze Units
4 Look Back
A rate of 44 ft/s is less than 50 ft/s.
A rate of 60 miles in 2 hours is 30
min/h or 0.5 mi/min.
Since 0.5 mi/min is less than 3000 ft/
60 s or 50 ft/s and 44 ft/s is less than
50 ft/s, then 44 ft/s is a reasonable
answer.
Pre-Algebra
7-3 Analyze Units
Try This: Example 3
A train traveled 180 miles on a railroad track
in 4 hours. How many feet per second was the
train traveling?
1
Understand the Problem
The problem is stated in units of miles and
hours. The question asks for the answer in
units of feet and seconds. You will need to use
several conversion factors.
List the important information:
5280 ft
• Miles to feet
1 mi
1h
• Hours to minutes
60 min
1 min
• Minutes to seconds
60 s
Pre-Algebra
7-3 Analyze Units
Try This: Example 3 Continued
2
Make a Plan
Multiply by each conversion factor
separately, or simplify the problem and
multiply by several conversion factors at
once.
Pre-Algebra
7-3 Analyze Units
3
Solve
First, convert 180 miles in 4 hours into a unit
rate.
180 mi = (180 ÷ 4) mi = 45 mi
4h
1h
(4 ÷ 4) h
Create a single conversion factor to convert
hours directly to seconds:
hours to minutes
1h
minutes to seconds
60 min
1 min
60 s
hours to seconds = 1 h • 1 min = 1 h
3600 s
60 min
60 s
45 mi •
1h
Pre-Algebra
5280 ft • 1 h
1 mi
3600 s
Set up the conversion factors.
7-3 Analyze Units
3 Solve
Do not include the numbers yet.
Notice what happens to the units.
mi • ft • h
mi s
h
Simplify. Only ft remains.
s
45 mi • 5280 ft • 1 h
Multiply.
3600 s
1 mi
1h
45 • 5280 ft • 1 = 237,600 ft = 66 ft
1 • 1 • 3600 s
3600 s
1s
The train was traveling 66 feet per second.
Pre-Algebra
7-3 Analyze Units
4 Look Back
A rate of 66 ft/s is more than 50 ft/s.
A rate of 180 miles in 4 hours is 45
mi/h or 0.75 mi/min.
Since 0.75 mi/min is more than 3000
ft/60 s or 50 ft/s and 66 ft/s is more
than 50 ft/s, then 66 ft/s is a
reasonable answer.
Pre-Algebra
7-3 Analyze Units
Additional Example 4: Physical Science Application
A strobe lamp can be used to measure the
1
speed of an object. The lamp flashes every 100
of a second. A camera records the object
moving 52 cm between flashes. How fast is
the object moving in m/s?
52 cm
1
s
100
Pre-Algebra
Use rate = distance .
time
7-3 Analyze Units
Additional Example 4 Continued
It may help to eliminate the fraction
52 cm = 100 • 52 cm
1
1 s
100
•
s
100
100
= 5200 cm
1s
Pre-Algebra
1
first.
100
Multiply top and
bottom by 100.
7-3 Analyze Units
Additional Example 4 Continued
Now convert centimeters to meters.
5200 cm
1s
= 5200 cm • 1 m
100 cm
1s
52 m
= 5200 m =
1s
100 s
The object is traveling 52 m/s.
Pre-Algebra
Multiply by the
conversion factor.
7-3 Analyze Units
Try This: Example 4
A strobe lamp can be used to measure the
1
speed of an object. The lamp flashes every 100
of a second. A camera records the object
moving 65 cm between flashes. How fast is
the object moving in m/s?
65 cm
1
s
100
Pre-Algebra
Use rate = distance .
time
7-3 Analyze Units
Try This: Example 4 Continued
It may help to eliminate the fraction
65 cm = 100 • 65 cm
1
1 s
100
•
s
100
100
= 6500 cm
1s
Pre-Algebra
1
first.
100
Multiply top and
bottom by 100.
7-3 Analyze Units
Try This: Example 4 Continued
Now convert centimeters to meters.
6500 cm
1s
= 6500 cm • 1 m
100 cm
1s
65 m
= 6500 m =
1s
100 s
The object is traveling 65 m/s.
Pre-Algebra
Multiply by the
conversion factor.
7-3 Analyze Units
Additional Example 5: Transportation Application
The rate 1 knot equals 1 nautical mile per hour.
One nautical mile is 1852 meters. What is the
speed in kilometers per hour of a ship traveling
at 5 knots?
5 knots = 5 nautical mi/h
km
Set up the units to obtain h in your answer.
m
nautical mi •
• km
Examine the units.
m
nautical
mi
h
5 nautical mi •
1852 m
• 1 km
1h
1 nautical mi 1000 m
= 5 • 1852 • 1 km = 9260 km = 9.26 km
1000 h
1h
1 h • 1 • 1000
The ship is traveling 9.26 km/h.
Pre-Algebra
7-3 Analyze Units
Try This: Example 5
The rate 1 knot equals 1 nautical mile per hour.
One nautical mile is 1852 meters. What is the
speed in kilometers per hour of a ship traveling
at 9 knots?
9 knots = 9 nautical mi/h
km
Set up the units to obtain h in your answer.
m
nautical mi •
• km
Examine the units.
m
nautical
mi
h
9 nautical mi •
1852 m
• 1 km
1h
1 nautical mi 1000 m
= 9 • 1852 • 1 km = 16,668 km  16.67 km
1000 h
1h
1 h • 1 • 1000
The ship is traveling about 16.67 km/h.
Pre-Algebra
7-3 Analyze Units
Lesson Quiz
Find the appropriate factor for each conversion.
1. kilograms to grams 1000 g
kg
2. pints to gallons 1 gal
8 pt
3. You drive 136 miles from your house to your aunt’s
house at the lake. You use 8 gallons of gas. How
many yards does your car get to the gallon? 29,920 yd
gal
4. A cheetah was timed running 200 yards in 6
seconds. What was the average speed in miles per
hour? ≈ 68 mi/h
Pre-Algebra