1. Pre-Algebra
7-3
Analyze Units
Learn to use one or more conversion factors to solve rate problems.

2. 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 unit.

3. 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.

4. Pre-Algebra
7-3
Analyze Units
Multiplying by a conversion factor is like multiplying by a fraction that reduces to
1, such as . 5
1 ft
12 in.
12 in. = = 1
1 ft
12 in.
1 ft = = 1

5. 7-3 Analyze Units Helpful Hint The conversion factor
Pre-Algebra
7-3
Analyze Units
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.
Helpful Hint

6. Example 1: Finding Conversion Factors
Pre-Algebra
7-3
Analyze Units
Example 1: Finding Conversion Factors
Find the appropriate factor for each conversion.
A. feet to yards
B. pounds to ounces
1 yd
3 ft
There are 3 feet in 1 yard. To convert feet to yards, multiply the number of feet by
16 oz
1 lb
There are 16 ounces in 1 pound. To convert pounds to ounces, multiply the number of pounds by

7. Example 2: Using Conversion Factors to Solve Problems
Pre-Algebra
7-3
Analyze Units
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
1 yr 12 mo
Multiply the ratio by the conversion factor
580 lb 12 mo =
Cancel yr units.
= 48.3 lb per month
Divide 580 by 12.

8. 7-3 Analyze Units Example 2 Continued
Pre-Algebra
7-3
Analyze Units
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.

9. 7-3 Analyze Units Try This: Example 2
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
Multiply the ratio by the conversion factor
23,000 mi 12 mo =
Cancel yr units.
= per month
Divide 23,000 by 12.
Sam drives his car about 1917 miles per month.

10. 7-3 Analyze Units Example 3: Problem Solving Application
Pre-Algebra
7-3
Analyze Units
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?
Understanding 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
1 mi
• Feet to miles
1 h
60 min
• Hours to minutes
1 min
60 s
• Minutes to seconds

11. 7-3 Analyze Units Solve First, convert 60 miles into feet.
Pre-Algebra
7-3
Analyze Units
Solve
First, convert 60 miles into feet.
5,280 ft
1 mi
60 mi =
316,800 ft •
Second, convert 2 hours into seconds.
minutes to seconds
1 min
60 s
hours to minutes
1 h
60 min
60 min
1 h
60 sec
1 min
2 hours • • =
7,200 sec
Then simplify,
316,800 ft
7,200 sec
44 ft
1 sec =

12. 7-3 Analyze Units Solve Do not include the numbers yet.
Pre-Algebra
7-3
Analyze Units
Solve
Do not include the numbers yet.
Notice what happens to the units. mi h • ft s
Simplify. Only remains. ft s •
60 mi
2 h
5280 ft
1 mi
1 h
3600 s
Multiply.
60 • 5280 ft • 1
2 • 1 • 3600 s =
316,800 ft
7200 s =
44 ft
1 s
The car was traveling 44 feet per second.

13. 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.

14. 7-3 Analyze Units Helpful Hint The conversion factor
Pre-Algebra
7-3
Analyze Units
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.
Helpful Hint

15. 7-3 Analyze Units Try This: Example 1
Pre-Algebra
7-3
Analyze Units
Try This: Example 1
Find the appropriate factor for each conversion.
A. minutes to seconds
B. hours to days
60 sec
1 min
There are 60 seconds in 1 minute. To convert minutes to seconds, multiply the number of minutes by
1 day
24 h
There are 24 hours in 1 day. To convert hours to days, multiply the number hours by

16. 7-3 Analyze Units Try This: Example 3
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?
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
1 mi
• Feet to miles
1 h
60 min
• Hours to minutes
1 min
60 s
• Minutes to seconds

17. 7-3 Analyze Units Solve Do not include the numbers yet.
Pre-Algebra
7-3
Analyze Units
Solve
Do not include the numbers yet.
Notice what happens to the units. mi h • ft s
Simplify. Only remains. ft s •
180 mi
4 h
5280 ft
1 mi
1 h
3600 s
Multiply.
180 • 5280 ft • 1
4 • 1 • 3600 s =
950,400 ft
14,400 s =
66 ft
1 s
The train was traveling 66 feet per second.

18. Example 4: Physical Science Application
Pre-Algebra
7-3
Analyze Units
Example 4: Physical Science Application
A strobe lamp can be used to measure the speed of an object. The lamp flashes every
of a second. A camera records the object moving 52 cm between flashes. How fast is the object moving in m/s? 1
100
distance .
time
Use rate =

19. 7-3 Analyze Units Multiply by the conversion factor.
Pre-Algebra
7-3
Analyze Units
Example 4 Continued
Now convert centimeters to meters.
52 cm
.01 s
52 cm
.01 s = •
1 m
100 cm
Multiply by the conversion factor.
52 m
1 s =
The object is traveling 52 m/s.

20. Example 5: Transportation Application
Pre-Algebra
7-3
Analyze Units
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
Set up the units to obtain in your answer. km h
nautical mi h • m km
Examine the units.
5 nautical mi
1 h •
1852 m
1 nautical mi
1 km
1000 m
5 • 1852 • 1 km
1 h • 1 • 1000 = =
9260 km
1000 h =
9.26 km
1 h
The ship is traveling 9.26 km/h.

21. 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.

22. 7-3 Analyze Units Helpful Hint The conversion factor
Pre-Algebra
7-3
Analyze Units
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.
Helpful Hint

23. 7-3 Analyze Units Try This: Example 4
Pre-Algebra
7-3
Analyze Units
Try This: Example 4
A strobe lamp can be used to measure the speed of an object. The lamp flashes every
of a second. A camera records the object moving 65 cm between flashes. How fast is the object moving in m/s? 1
100
distance .
time
Use rate =

24. Try This: Example 4 Continued
Pre-Algebra
7-3
Analyze Units
Try This: Example 4 Continued
Now convert centimeters to meters.
65 cm
.01 s
65 cm
.01 s = •
1 m
100 cm
Multiply by the conversion factor.
65 m
1 s =
The object is traveling 65 m/s.

25. 7-3 Analyze Units Try This: Example 5
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
Set up the units to obtain in your answer. km h
nautical mi h • m km
Examine the units.
9 nautical mi
1 h •
1852 m
1 nautical mi
1 km
1000 m
9 • 1852 • 1 km
1 h • 1 • 1000 = =
16,668 km
1000 h ≈
16.67 km
1 h
The ship is traveling about km/h.

26. Example 4: Physical Science Application
Pre-Algebra
7-3
Analyze Units
Example 4: Physical Science Application
A strobe lamp can be used to measure the speed of an object. The lamp flashes every
of a second. A camera records the object moving 52 cm between flashes. How fast is the object moving in m/s? 1
100
distance .
time
Use rate =