Analyze Units

7.3

Pre-Algebra

Learn

to use one or more conversion factors to solve rate problems.

Vocabulary conversion factor

Dimensional Analysis

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.

Conversion Factors

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

Conversion Factors

Multiplying by a conversion factor is like multiplying by a fraction that reduces to 1, such as .

5 1 ft 12 in.

= 12 in.

12 in.

, or 1 ft 1 ft = 1

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.

Example: 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

B. pounds to ounces

feet by .

There are 16 ounces in 1 pound. To convert pounds to ounces, multiply the number of pounds 16 oz 1 lb

Try This

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

B. hours to days

60 sec 1 min There are 24 hours in 1 day. To convert hours to days, multiply the number hours 1 day 24 h .

by

Example: 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.

Example 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.

Try This

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 = 23,000 mi 12 mo 1 yr 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.

Example: 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: • Miles to feet 5280 ft 1 mi • Hours to minutes • Minutes to seconds 1 h 60 min 1 min 60 s

Example Continued

2 Make a Plan

Multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.

Example Continued

3 Solve

First, convert 60 miles in 2 hours into a unit rate.

60 mi 2 h = (60÷2) mi (2÷2) h = 30 mi 1 h Create a single conversion factor to convert hours directly to seconds: hours to minutes 1 h 60 min minutes to seconds 1 min 60 s hours to seconds = 1 h 60 min 30 mi 1 h • 5280 ft 1 mi • 1 h 3600 s • 1 min 60 s = 1 h 3600 s

Set up the conversion factors.

Example Continued

3 Solve

Do not include the numbers yet.

Notice what happens to the units.

mi h • ft mi • h s 30 mi 1 h • 5280 ft 1 mi • 1 h 3600 s

Multiply.

30 • 5280 ft • 1 1 • 1 • 3600 s = 158,400 ft The car was traveling 44 feet per second.

Example Continued

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.

Try This

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: • Miles to feet 5280 ft 1 mi • Hours to minutes • Minutes to seconds 1 h 60 min 1 min 60 s

Try This Continued

2 Make a Plan

Multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.

Try This Continued

3 Solve

First, convert 180 miles in 4 hours into a unit rate.

180 mi 4 h = (180 ÷ 4) mi (4 ÷ 4) h = 45 mi 1 h Create a single conversion factor to convert hours directly to seconds: hours to minutes 1 h 60 min minutes to seconds 1 min 60 s hours to seconds = 1 h 60 min 45 mi 1 h • 5280 ft 1 mi • 1 h 3600 s • 1 min 60 s = 1 h 3600 s

Set up the conversion factors.

Try This Continued

3 Solve

Do not include the numbers yet.

Notice what happens to the units.

mi h • ft mi • h s 45 mi 1 h • 5280 ft 1 mi • 1 h 3600 s

Multiply.

45 • 5280 ft • 1 1 • 1 • 3600 s = 237,600 ft The train was traveling 66 feet per second.

Try This Continued

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.

Example:

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?

52 cm 1 100 s

Use rate = distance .

time

Example Continued

52 cm 1 100 s = 100 • 52 cm 100 • 100 s = 5200 cm 1 s

Multiply top and bottom by 100.

Example Continued

Now convert centimeters to meters.

= 5200 cm 1 s 5200 cm 1 s = 5200 m 100 s = • 1 m 100 cm 52 m 1 s

Multiply by the conversion factor.

The object is traveling 52 m/s.

Try This

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?

65 cm 1 100 s

Use rate = distance .

time

Try This Continued

65 cm 1 100 s = 100 • 65 cm 100 • 100 s = 6500 cm 1 s

Multiply top and bottom by 100.

Try This Continued

Now convert centimeters to meters.

= 6500 cm 1 s 6500 cm 1 s = 6500 m 100 s = • 1 m 100 cm 65 m 1 s

Multiply by the conversion factor.

The object is traveling 65 m/s.

Example:

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 nautical mi h 5 nautical mi = 1 h • • 5 • 1852 • 1 km 1 h • 1 • 1000 m nautical mi • km h km m 1852 m 1 nautical mi •

Examine the units.

1 km 1000 m The ship is traveling 9.26 km/h.

Try This

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 nautical mi h 9 nautical mi = 1 h • • 9 • 1852 • 1 km 1 h • 1 • 1000 m nautical mi • km h km m 1852 m 1 nautical mi •

Examine the units.

1 km 1000 m  16.67 km 1 h The ship is traveling about 16.67 km/h.

Lesson Quiz

Find the appropriate factor for each conversion.

1. kilograms to grams 2. pints to gallons 1 gal 8 pt 1000 g kg 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