Linear Measurement

The U.S. system of measurement uses the inch, foot, yard, and mile to measure length.

U.S. Units of Length

12 inches (in.) = 1 foot (ft)

3 feet = 1 yard (yd)

36 inches = 1 yard

5280 feet = 1 mile (mi)

Unit Fractions

12 in. 1 ft

11ft 12 in.

3 ft 1yd

11yd 3 ft

5280 ft 1mi

11mi 5280 ft

To convert from one unit of length to another,To convert from one unit of length to another, unit fractions may be unit fractions may be used. A unit fraction is a fraction that equals 1.used. A unit fraction is a fraction that equals 1.

To convert 60 inches to feet,To convert 60 inches to feet, multiply by a unit fractionmultiply by a unit fraction that that relates feet to inches. The unit fraction should be written so relates feet to inches. The unit fraction should be written so thatthat the units we are converting to,the units we are converting to, feet,feet, are in the numerator are in the numerator and the original units,and the original units, inches,inches, are in the denominator.are in the denominator.

To convert 60 inches to feet,To convert 60 inches to feet, multiply by a unit fractionmultiply by a unit fraction that that relates feet to inches. The unit fraction should be written so relates feet to inches. The unit fraction should be written so thatthat the units we are converting to,the units we are converting to, feet,feet, are in the numerator are in the numerator and the original units,and the original units, inches,inches, are in the denominator.are in the denominator.

Unit fractionunits converting to

original units

60 in. 1 ft

1 12 in.60 in.

60 1ft

1 12

60 ft5 ft

12

The basic unit of length in the metric system is the meter. A meter is slightly longer than a yard. It is approximately 39.37 inches long. Like the decimal system, the metric system uses powers of ten to define units.

Metric System of Measurement

Metric System of Measurement

Prefix kilo hecto deka

deci centi milli

Meaning 1000 100 10

1/10 1/100 1/1000

1 millimeter (mm) 1/1000 or 0.001 m

1 centimeter (cm) 1/100 or 0.01 m

1 decimeter (dm) 1/10 or 0.1 m

1 meter (m) 1 m 1 dekameter (dam) 10 m 1 hectometer (hm) 100 m

1 kilometer (km) 1000 meters (m)

Metric Unit of Length

Metric System of Measurement

The most commonly used measurements of length in the metric system are the meter,

millimeter, centimeter, and kilometer.

Metric System of Measurement

As with the U.S. system of measurement, unit fractions may be used to convert from one unit of

length to another. The major advantage of the metric system is the ease of converting from one unit of

length to another. Since all units of length are powers of 10 of the meter, converting from one unit

of length to another is as simple as moving the decimal point.

Converting Between Measurements

Listing units of length in order from largest to smallest helps keep track of how many places to

move the decimal point when converting.

km hm dam m dm cm mm

Using the listing of units of length, convert 3.5 m to centimeters.Using the listing of units of length, convert 3.5 m to centimeters.

Start End

2 units to the right

3.50 m = 350. cm or 350 cm

2 places to the right

Converting Between Measurements

The process of converting from one unit to another is called dimensional analysis, or unit analysis. To convert units, multiply by one or more conversion factors. A conversion factor is a ratio of two quantities that are equal but use different units.

For example, to convert inches to feet

you would use the ratio as a conversion factor.1 ft12 in.

Multiplying by a conversion factor is like multiplying by 1.

12 in.12 in.

1 ft1 ft

= = = 11 ft12 in.

The average American uses 580 pounds of paper per year. Find this rate in pounds per month, to the nearest tenth.

Additional Example 1: Using Conversion Factors to Solve Problems

Convert the rate 580 pounds per year to pounds per month.

580 lb 1 yr

1 yr 12 mo

580 lb 12 mo

48.3 lb per month

To convert the second amount in a rate, multiply by the conversion factor with that unit in the first quantity.

Divide 580 by 12.

The average American uses 48.3 pounds of paper per month.

Divide out like units. •yrmo =

lbmo

lb yr

Check It Out! Example 1

Sam drives his car 23,040 miles per year. Find this rate in the number of miles driven per month, to the nearest mile.

Convert the rate 23,040 miles per year to miles per month.

23,040 mi 1 yr

1 yr 12 mo

23,040 mi 12 mo

=

= 1920 per month Divide 23,040 by 12.

Sam drives his car 1920 miles per month.

Divide out units. •yrmo

= mimo

miyr

To convert the second amount in a rate, multiply by the conversion factor with that unit in the first quantity.

Additional Example 2: Problem Solving Application

A car traveled 60 miles on a road in 2 hours. Find this rate in feet per second.

11 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:

• Feet to miles5280 ft

1 mi

• Hours to minutes

• Minutes to seconds 1 min60 s

1 h60 min

You know the conversion factor that converts miles to feet. So multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.

22 Make a Plan

Convert to miles per hour.

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

Solve Continued33

30 • 5280 ft • 1 1 • 1 • 3600 s

= 158,400 ft3600 s

= 44 ft1 s

The car was traveling 44 feet per second.

Divide out like units.• •30 mi1 h

5280 ft1 mi

1 h 3600 s=

=

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.

Check It Out! Example 2

A train traveled 180 miles on a railroad track in 4 hours. Find this rate in feet per second.

11 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:

• Feet to miles5280 ft

1 mi

• Hours to minutes

• Minutes to seconds 1 min60 s

1 h60 min

22 Make a Plan

You know the conversion factor that converts miles to feet. So multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.

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

Solve Continued33

45 • 5280 ft • 1 1 • 1 • 3600 s

= 237,600 ft3600 s

= 66 ft1 s

The train was traveling 66 feet per second.

Divide out like units.• •45 mi1 h

5280 ft1 mi

1 h 3600 s

=

=

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.

Additional Example 3: 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? Use dimensional analysis to check the reasonableness of your answer.

1 100

distance .time

Use rate =52 cm

1100

s

It may help to eliminate the fraction first.

Additional Example 3 Continued

1 100

Multiply numerator and denominator by 100.

5200 cm1 s

=

52 cm1

100s

= 100 • 52 cm 1

100 s 100 •

Convert centimeters to meters to see if the answer is reasonable.

Additional Example 3 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

Check It Out! Example 3

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

It may help to eliminate the fraction first.

Check It Out! Example 3 Continued

1 100

Multiply numerator and denominator by 100.

6500 cm1 s

=

65 cm1

100s

= 100 • 65 cm 1

100 s 100 •

Convert centimeters to meters to see if the answer is reasonable.

Check It Out! Example 3 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

Lesson Quiz

1. 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?

2. A cheetah was timed running 200 yards in 6 seconds. What was its

average speed in miles per hour?

29,920 ydgal

≈ 68 mi/h

Convert from: To: Multiply by:

mile kilometer (km)1.609347

inch millimeter (mm) 25.4

inch centimeter (cm) 2.54

foot meter (m) 0.3048

yard meter (m) 0.9144

Which of the following represents the longest length?

 

A. 248 mm

B. 21 cm

C. 2 m

D. 0.9 km

Answer:

0.9 km

Convert half of a mile, three yards, and two feet into total feet.

Answer:

2651 feet

Mark has three 650-milliliter bottles of lemonade. Is one 2-liter bottle large enough to hold all of the lemonade from the three bottles? How much

more or less would there be?

Answer:

Yes.(Total is 1950ml = 1.95 liters)

50 ml more room left in 2 liter bottle.