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

1
1 ft
12 in.
3 ft 1 yd

1
1 yd 3 ft
5280 ft
1mi

1
1mi
5280 ft
To convert from one unit of length to another, unit fractions may be
used. A unit fraction is a fraction that equals 1.
To convert 60 inches to feet, multiply by a unit fraction that
relates feet to inches. The unit fraction should be written so
that the units we are converting to, feet, are in the numerator
and the original units, inches, are in the denominator.
Unit fraction
units converting to
60 in. 1 ft
60  1 ft 60 ft
60 in. 



 5 ft
1
12 in.
1  12
12
original units
Metric System of Measurement
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
Meaning
1000
100
10
deci
centi
milli
1/10
1/100
1/1000
Metric Unit of Length
1 kilometer (km)  1000 meters (m)
1 hectometer (hm)  100 m
1 dekameter (dam)  10 m
1 meter (m)  1 m
1 decimeter (dm)  1/10 or 0.1 m
1 centimeter (cm)  1/100 or 0.01 m
1 millimeter (mm)  1/1000 or 0.001 m
Metric System of Measurement
The most commonly used measurements of
length in the metric system are the meter,
millimeter, centimeter, and kilometer.
Converting Between Measurements
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.
Start
km
hm
dam
End
m
dm
cm
mm
2 units to the right
Using the listing of units of length, convert 3.5 m to centimeters.
3.50 m = 350. cm or 350 cm
2 places to the right
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 1 ft as a conversion factor.
12 in.
Multiplying by a conversion factor is like
multiplying by 1.
1 ft = 12 in.
12 in.
12 in.
= 1 ft = 1
1 ft
Additional Example 1: Using Conversion Factors to
Solve Problems
The average American uses 580 pounds of paper per year. Find this
rate in pounds per month, to the nearest tenth.
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.
yr
lb
lb
=
•
Divide out like units.
yr mo mo
Divide 580 by 12.
The average American uses 48.3 pounds of paper per month.
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
=
23,040 mi 12
mo
1 yr 12
mo
To convert the second amount in a rate,
multiply by the conversion factor with
that unit in the first quantity.
Divide out units.
mi • yr = mi
yr mo mo
Divide 23,040 by 12.
= 1920 per month
Sam drives his car 1920 miles per month.
Additional Example 2: Problem Solving Application
A car traveled 60 miles on a road in 2 hours. Find
this rate in feet per second.
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:
• Feet to miles
• Hours to minutes
• Minutes to seconds
5280 ft
1 mi
1h
60 min
1 min
60 s
2
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.
3
Solve
60 mi =
2h
(60÷2) mi
(2÷2) h
=
30 mi Convert to miles per hour.
1h
Create a single conversion factor to convert hours
directly to seconds:
1h
60 min
hours to minutes
1h
•
60 min
hours to seconds =
30 mi
1h
•
5280 ft
1 mi
minutes to seconds
•
1 min =
60 s
1 min
60 s
1h
3600 s
1h
Set up the conversion factors.
3600 s
Solve Continued
3
5280 ft • 1 h
30
mi
•
=
3600 s
1 mi
1h
=
30 • 5280 ft • 1
1 • 1 • 3600 s
=
Divide out like units.
158,400 ft = 44 ft
3600 s
1s
The car was traveling 44 feet per second.
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 mi/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.
Check It Out! Example 2
A train traveled 180 miles on a railroad track in
4 hours. Find this rate in feet per second.
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:
5280 ft
• Feet to miles
1 mi
1h
• Hours to minutes
60 min
• Minutes to seconds
1 min
60 s
2
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.
3
Solve
First, convert 180 miles in 4 hours into a unit rate.
180 mi =
4h
(180 ÷ 4) mi
(4 ÷ 4) h
= 45 mi
1h
Create a single conversion factor to convert hours
directly to seconds:
1h
60 min
hours to minutes
1h
•
60 min
hours to seconds =
45 mi
1h
•
5280 ft
1 mi
minutes to seconds
•
1 min =
60 s
1 min
60 s
1h
3600 s
1h
Set up the conversion factors.
3600 s
3
Solve Continued
1h
5280
ft
45
mi
•
•
Divide out like units.
=
3600 s
1 mi
1h
45 • 5280 ft • 1 = 237,600 ft = 66 ft
=
1 • 1 • 3600 s
3600 s
1s
The train was traveling 66 feet per second.
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.
Additional Example 3: Physical Science Application
A strobe lamp can be used to measure the speed of an object.
1
The lamp flashes every
100
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.
52 cm
1
s
100
Use rate =
distance .
time
Additional Example 3 Continued
It may help to eliminate the fraction
52 cm = 100 • 52 cm
1
100 • 1 s
s
100
100
= 5200 cm
1s
1
first.
100
Multiply numerator and
denominator by 100.
Additional Example 3 Continued
Convert centimeters to meters to see if the answer is reasonable.
5200 cm
1s
1m
= 5200 cm •
100 cm
1s
52 m
5200
m
=
=
1s
100 s
The object is traveling 52 m/s.
Multiply by the
conversion factor.
Check It Out! Example 3
A strobe lamp can be used to measure the speed of an object.
1
The lamp flashes every
100
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
s
100
Use rate =
distance .
time
Check It Out! Example 3 Continued
It may help to eliminate the fraction
65 cm = 100 • 65 cm
1
100 • 1 s
s
100
100
= 6500 cm
1s
1
first.
100
Multiply numerator and
denominator by 100.
Check It Out! Example 3 Continued
Convert centimeters to meters to see if the answer is reasonable.
6500 cm
1s
1m
= 6500 cm •
100 cm
1s
65 m
6500
m
=
=
1s
100 s
The object is traveling 65 m/s.
Multiply by the
conversion factor.
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?
29,920 yd
gal
2. A cheetah was timed running 200 yards in 6 seconds. What was its
average speed in miles per hour?
≈ 68 mi/h
Convert from:
mile
inch
inch
foot
yard
To:
Multiply by:
kilometer (km)
1.609347
millimeter (mm)
25.4
centimeter (cm)
2.54
meter (m)
0.3048
meter (m)
0.9144
Which of the following
represents the longest length?
A.
B.
C.
D.
248 mm
21 cm
2m
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.