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 =