pre-algebra 7-3 analyze units learn to use one or more conversion factors to solve rate problems
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Pre-Algebra
7-3 Analyze Units
Learn to use one or more conversion factors to solve rate problems.
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.
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 ft12 in.
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
Pre-Algebra
7-3 Analyze Units
Find the appropriate factor for each conversion.
Example 1: Finding Conversion Factors
A. feet to yards
B. pounds to ounces
1 yd3 ft
There are 3 feet in 1 yard. To convert feet to
yards, multiply the number of feet by .
16 oz1 lb
There are 16 ounces in 1 pound. To convert
pounds to ounces, multiply the number of
pounds by .
Pre-Algebra
7-3 Analyze Units
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.
Example 2: Using Conversion Factors to Solve Problems
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
580 lb 12 mo
=
= 48.3 lb per month
Multiply the ratio by the conversion factor
Cancel yr units.
Divide 580 by 12.
Pre-Algebra
7-3 Analyze Units
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.
Example 2 Continued
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
Understanding The Problem
Example 3: Problem Solving ApplicationA car traveled 60 miles on a road in 2 hours. How many feet per second was the car traveling?
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:
• Feet to miles5280 ft
1 mi• Hours to minutes
• Minutes to seconds 1 min60 s
1 h60 min
Pre-Algebra
7-3 Analyze Units
First, convert 60 miles into feet.
60 mi 5,280 ft1 mi
= 316,800 ft
Second, convert 2 hours into seconds.
2 hours • 60 sec1 min
minutes to seconds 1 min60 s
hours to minutes 1 h60 min
60 min1 h
Solve
316,800 ft7,200 sec
=
•
• 7,200 sec=
Then simplify, 44 ft1 sec
Pre-Algebra
7-3 Analyze Units
Solve
Do not include the numbers yet.
Notice what happens to the units.
60 • 5280 ft • 1 2 • 1 • 3600 s
= 316,800 ft7200 s
= 44 ft1 s
The car was traveling 44 feet per second.
Simplify. Only remains.fts
Multiply.
mih
•ftmi
• hs
• •60 mi2 h
5280 ft1 mi
1 h 3600 s
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 ft12 in.
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
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 sec1 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
.
Pre-Algebra
7-3 Analyze UnitsTry This: Example 3
A train traveled 180 miles on a railroad track in 4 hours. How many feet per second was the train traveling?
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:
• Feet to miles5280 ft
1 mi• Hours to minutes
• Minutes to seconds 1 min60 s
1 h60 min
Understand the Problem
Pre-Algebra
7-3 Analyze Units
Solve
Do not include the numbers yet.
Notice what happens to the units.
180 • 5280 ft • 1 4 • 1 • 3600 s
= 950,400 ft14,400 s
= 66 ft1 s
The train was traveling 66 feet per second.
Simplify. Only remains.fts
Multiply.
mih
•ftmi
• hs
• •180 mi4 h
5280 ft1 mi
1 h 3600 s
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 =
Pre-Algebra
7-3 Analyze Units
Now convert centimeters to meters.
Example 4 Continued
52 cm.01 s
Multiply by the conversion factor.
52 m1 s
=
The object is traveling 52 m/s.
52 cm.01 s
= • 1 m100 cm
Pre-Algebra
7-3 Analyze Units
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?
Example 5: Transportation Application
5 knots = 5 nautical mi/h
Set up the units to obtain in your answer.kmh
Examine the units.
5 • 1852 • 1 km1 h • 1 • 1000
= = 9260 km1000 h
= 9.26 km1 h
The ship is traveling 9.26 km/h.
nautical mih
• mnautical mi
• kmm
5 nautical mi1 h
• •1852 m1 nautical mi
1 km1000 m
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 ft12 in.
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
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 =
Pre-Algebra
7-3 Analyze Units
Now convert centimeters to meters.
Try This: Example 4 Continued
65 cm.01 s
Multiply by the conversion factor.
65 m1 s
=
The object is traveling 65 m/s.
65 cm.01 s
= • 1 m100 cm
Pre-Algebra
7-3 Analyze Units
Try This: Example 5The 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.kmh
Examine the units.
9 • 1852 • 1 km1 h • 1 • 1000
= =16,668 km1000 h
≈ 16.67 km1 h
The ship is traveling about 16.67 km/h.
nautical mih
• mnautical mi
• kmm
9 nautical mi1 h
• •1852 m1 nautical mi
1 km1000 m
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 =
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