Pre-Algebra

7-3 Analyze Units7-3 Analyze Units

Pre-Algebra

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Pre-Algebra

7-3 Analyze Units

Warm UpFind each unit rate.

1. Jump rope 192 times in 6 minutes

2. Four pounds of bananas for $2.36

3. 16 anchor bolts for $18.56

4. 288 movies on 9 shelves

32 jumps/min

$0.59/lb

$1.16/bolt

32 movies/shelf

Pre-Algebra

7-3 Analyze Units

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. ••••

•••

= Possible answer:1365

420

=

Pre-Algebra

7-3 Analyze Units

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 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

Multiplying by a conversion factor is like multiplying by a fraction that reduces to 1, such as .5

5

12 in.12 in.

1 ft1 ft

= , or = 11 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.

Additional 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

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 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.

Additional 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.

Additional 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

11 Understand the Problem

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?

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

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

22 Make a Plan

Additional Example 3 Continued

Pre-Algebra

7-3 Analyze Units

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

Solve33

60 mi2 h

= (60÷2) mi(2÷2) h

= 30 mi1 h

Create a single conversion factor to convert hours directly to seconds:

hours to seconds = • 1 min60 s

Set up the conversion factors.

minutes to seconds 1 min60 s

hours to minutes 1 h60 min

1 h60 min

1 h3600 s

=

30 mi1 h

• 5280 ft1 mi

• 1 h 3600 s

Pre-Algebra

7-3 Analyze Units

Solve33

Do not include the numbers yet.

Notice what happens to the units.

30 • 5280 ft • 1 1 • 1 • 3600 s

= 158,400 ft3600 s

= 44 ft1 s

The car was traveling 44 feet per second.

Simplify. Only remains.fts

Multiply.

mih

•ftmi

• hs

• •30 mi1 h

5280 ft1 mi

1 h 3600 s

Pre-Algebra

7-3 Analyze Units

A rate of 44 ft/s is less than 50 ft/s. A rate of 60 miles in 2 hours is 30 mi/h or 0.5 mi/min.

44 Look Back

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

11 Understand the Problem

Try This: Example 3A 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

Pre-Algebra

7-3 Analyze Units

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

22 Make a Plan

Try This: Example 3 Continued

Pre-Algebra

7-3 Analyze Units

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

Solve33

180 mi4 h

= (180 ÷ 4) mi(4 ÷ 4) h

= 45 mi1 h

Create a single conversion factor to convert hours directly to seconds:

hours to seconds = • 1 min60 s

Set up the conversion factors.

minutes to seconds 1 min60 s

hours to minutes 1 h60 min

1 h60 min

1 h3600 s

=

45 mi1 h

• 5280 ft1 mi

• 1 h 3600 s

Pre-Algebra

7-3 Analyze Units

Solve33

Do not include the numbers yet.

Notice what happens to the units.

45 • 5280 ft • 1 1 • 1 • 3600 s

= 237,600 ft3600 s

= 66 ft1 s

The train was traveling 66 feet per second.

Simplify. Only remains.fts

Multiply.

mih

•ftmi

• hs

• •45 mi1 h

5280 ft1 mi

1 h 3600 s

Pre-Algebra

7-3 Analyze Units

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.

44 Look Back

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 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 =52 cm1

100s

Pre-Algebra

7-3 Analyze Units

It may help to eliminate the fraction first.

Additional Example 4 Continued

1 100

Multiply top and bottom by 100.

5200 cm1 s

=

52 cm1

100s

= 100 • 52 cm 1

100 s 100 •

Pre-Algebra

7-3 Analyze Units

Now convert centimeters to meters.

Additional Example 4 Continued

5200 cm1 s

Multiply by the conversion factor.

5200 m100 s

=52 m1 s

=

The object is traveling 52 m/s.

5200 cm1 s

= • 1 m100 cm

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 =65 cm1

100s

Pre-Algebra

7-3 Analyze Units

It may help to eliminate the fraction first.

Try This: Example 4 Continued

1 100

Multiply top and bottom by 100.

6500 cm1 s

=

65 cm1

100s

= 100 • 65 cm 1

100 s 100 •

Pre-Algebra

7-3 Analyze Units

Now convert centimeters to meters.

Try This: Example 4 Continued

6500 cm1 s

Multiply by the conversion factor.

6500 m100 s

=65 m1 s

=

The object is traveling 65 m/s.

6500 cm1 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?

Additional 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

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.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

Lesson Quiz

Find the appropriate factor for each conversion.

1. kilograms to grams

2. pints to gallons

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?

4. A cheetah was timed running 200 yards in 6

seconds. What was the average speed in miles per

hour?

1000 gkg

1 gal8 pt

29,920 ydgal

≈ 68 mi/h