PERFORM ARITHMETIC CALCULATIONS ON UNITS
OF LENGTH, WEIGHT, AND CAPACITY
Study Notes
Measurement
There are two major systems of measurement in use today. They are the
United States system and the metric system.
The metric system of measurement is the decimal system of measure
developed in the 1790s in France. This system which is also referred to as SI
is the official unit used by every country except the United States. These
units use a standard set of prefixes such as milli, centi, kilo, deci
The US Customary Units is a system of measurement primarily used in the
United States which was validated in 1832. This system of measurement
came from the English units, the units of measure used in the British Empire
before the formulation of the Imperial system on January 1, 1826, and before
American Independence. The customary units are widely used in the areas of
commerce and industry. Consumer products and industrial manufacturing
materials are weighed or measured using the customary units as evident in
their labels.
The common customary units of measure used are:
inch, foot, yard, and mile for lengths
acre for area
cubic inch, cubic foot, and cubic yard for volume and capacity
ounce, pint, quart, and gallon for fluid volume
pint and quart for dry volume
This section begins by defining measurement and its common uses.
1.1.1: Measurement and the United States System
6.1.1 Learning Objectives
Define measurement.
Perform arithmetic calculations on units of length, weight, and capacity.
Convert from one unit of measure in the United States system to another
unit of measure.
Solve application problems using each form of measurement.
Measurement
Before we work the United States measurement system and the metric
system, let's gain a clear understanding of the concept of measurement.
Definition: Measurement
Measurement is comparison to some standard.
Definition: Standard Unit of Measure
The concept of measurement is based on the idea of direct comparison. This
means that measurement is the result of the comparison of two quantities.
The quantity that is used for comparison is called the standard unit of
measure.
Over the years, standards have changed. Quite some time in the past, the
standard unit of measure was determined by a king. For example,
1 inch was the distance between the tip of the thumb and the knuckle of the
king.
1 inch was also the length of 16 barley grains placed end to end.
Today, standard units of measure rarely change. Standard units of measure
are the responsibility of the Bureau of Standards in Washington D.C.
Some desirable properties of a standard are the following:
Accessibility. We should have access to the standard so we can make
comparisons.
Invariance. We should be confident that the standard is not subject to
change.
Reproducibility. We should be able to reproduce the standard so that
measurements are convenient and accessible to many people.
Definition: Length, Weight, Capacity
Length is a linear measurement to describe the distance between two
objects.
Weight is used to describe how heavy or light an object is.
Capacity is used to describe the amount of liquid (or other pourable
substance) that will fill an object.
The United States System of Measurement
Some of the common units (along with their abbreviations) for the United
States system of measurement are listed in the following table.
Unit Conversion Table
Length
Weight
1 foot (ft) = 12 inches (in)
1 yard (yd) = 3 feet (ft)
1 mile (mi) = 5,280 feet
1 pound (lb) =16 ounces (oz)
1 ton (T) = 2,000 pounds
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1 tablespoon (tbsp) = 3 teaspoons (tsp)
1 fluid ounce (fl oz) = 2 tablespoons
Liquid Capacity 1 cup (c) = 8 fluid ounces 1 pint (pt) = 2 cups
1 quart (qt) = 2 pints
1 gallon (gal) = 4 quarts
Conversions in the United States System
It is often convenient or necessary to convert from one unit of measure to
another. For example, it may be convenient to convert a measurement of
length that is given in feet to one that is given in inches. Such conversions
can be made using unit fractions.
Definition: Unit Fraction
A unit fraction is a fraction with a value of 1 in the numerator or the
denominator.
Unit fractions are formed by using two equal measurements. One
measurement is placed in the numerator of the fraction, and the other in the
denominator. Placement depends on the desired conversion.
Placement of Units
Place the unit being converted to in the numerator.
Place the unit being converted from in the denominator.
For example,
Equal Measurements Unit Fraction
1 ft = 12 in or
1 pt = 16 fl oz 1 pt or
1 pt
1 wk = 7 da or
Example 1
Make the following conversions. If a fraction occurs, convert it to a decimal
rounded to two decimal places.
Convert 11 yards to feet.
Solution
Looking in the unit conversion table under length, we see that . There are
two corresponding unit fractions, 1 yd
1 yd = 3 ft
and . Which one should we use? Look to see which unit we wish to
convert to. Choose the unit fraction with this unit in
1 yd
the numerator. We will choose 3 ft since this unit fraction has feet in
the numerator. Now, multiply 11 yd by the unit
1 yd
fraction. Notice that since the unit fraction has the value of 1, multiplying by
it does not change the value of 11 yd.
11 yd
11 yd ⋅ 1 yd
11 yd 3 ft
1⋅(Units can be added, subtracted,
multiplied, and divided, just as numbers can.)
1 yd
33 ft
Thus, .
11 yd = 33 ft
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Example 2
Convert 36 fl oz to pints.
Solution
Looking in the unit conversion table under liquid volume, we see that .
Since we are to convert to pints, we will
construct a unit fraction with pints in the numerator. 1 pt = 16 fl oz
1 pt
⋅
36 fl oz 1 pt
⋅16 fl oz
36 ⋅ 1 pt
16
36 pt
16
pt Convert to decimals: = 2.25
Thus, .
36 fl oz = 2.25 pt
Example 5
Your want to lay out a red carpet walkway for your grandma's 75th birthday.
The distance from the dropoff area to the entrance of the party venue is 18
feet. The fabric you want is only sold in yards. How many yards of fabric do
you need?
Solution
The unit fraction needed for converting from feet to yards is 1 yd .
18 ft 1 yd
18 ft ⋅Divide out common units.
1 3 ft
1 yd
1 ⋅ 3
18 yd
1 ⋅ 3
6 yd
Thus, . You would need to purchase 6 yards of fabric for the red carpet.
18 ft= 6 yd
Try It Now 1
Make the following conversions. If a fraction occurs, convert it to a decimal
rounded to two decimal places.
Convert 2 mi to feet.
Convert 26 ft to yards.
Convert 9 qt to pints.
Answer
10,560 ft
8.67 yd.
18 pt
This page titled 1.1.1: Measurement and the United States System is shared
under a CC BY license and was authored, remixed, and/or curated by Leah
Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
1.1.1.3 146699
1.1.1.1: Exercises
In the U.S. system, abbreviations can be difficult to associate with the
correct unit of measurement. What unit do these abbreviations belong to?
lb
oz
ft
mi
yd f. pt
qt
Convert the following:
52 inches to feet
257 miles to inches
11 feet to yards
47 ounces to pounds
2.3 tons to ounces f. 7 gallons to quarts
10 cups to quarts
3.6 quarts to cups
29 cups to gallons
3 pints to fluid ounces
417 ounces to ton
1 gallons to cups
8 2
2 mi to inches
3 cups to tablespoons
16 tsp to cups
A recipe calls for 3 teaspoons of soy sauce. You have a 5 pt container.
How many teaspoons could you get from your container of soy sauce?
The coach of a local sports team brought 6 gallons of water that needs
to be put into bottles that only hold one pint each. How many bottles will
they be able to fill?
An 8 ft length of 4 inch wide crown molding costs $14. How much will it
cost to buy 40 ft of crown molding?
A chain weighs 10 pounds per foot. How many ounces will 4 inches
weigh?
Sugar contains 15 calories per teaspoon. How many calories are in 1
cup of sugar?
The store is selling lemons at 2 for $1. Each yields about 2 tablespoons
of juice. How much will it cost to buy enough lemons to
make a 9-inch lemon pie requiring cup of lemon juice?
2
This page titled 1.1.1.1: Exercises
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1.1.2: The Metric System of Measurement
6.1.2 Learning Objectives
Describe some of the advantages of the base ten number system
Define prefixes of the metric measures
Convert from one unit of measure in the metric system to another unit of
measure
The Advantages of the Base Ten Number System
The metric system of measurement takes advantage of our base ten number
system. The advantage of the metric system over the United States system
is that in the metric system it is possible to convert from one unit of measure
to another simply by multiplying or dividing the given number by a power of
10. This means we can make a conversion simply by moving the decimal
point to the right or the left.
Prefixes
Common units of measure in the metric system are the meter (for length),
the liter (for capacity), and the gram (for mass). The units are abbreviated m
for meter, L for liter, and g for gram. To each of the units a prefix can be
attached. The metric prefixes along with their meaning are listed below.
Metric Prefixes and Abbreviations
giga (g): billion
mega (m): million
kilo (k): thousand
hecto (h): hundred
deka (dk): ten
deci (d): tenth
centi (c): hundredth
milli (m): thousandth
For example, if length is being measured,
1 kilometer is equivalent to 1000 meters. 1 km = 1000 m
1 centimeter is equivalent to one hundredth of a meter. 1 cm = 0.01 m
1 millimeter is equivalent to one thousandth of a meter. 1 mm = 0.001 m
Conversion from One Unit to Another Unit
Let's note three characteristics of the metric system that occur in the metric
table of measurements.
In each category, the prefixes are the same.
We can move from a larger to a smaller unit of measure by moving the
decimal point to the right. (or multiplying by 10 for each step)
We can move from a smaller to a larger unit of measure by moving the
decimal point to the left. (or dividing by 10 for each step)
The following table provides a summary of the relationship between the
basic unit of measure (meter, gram, liter) and each prefix, and how many
places the decimal point is moved and in what direction.
kilo hecto deka unit deci centi milli
Basic Unit to Prefix
unit to deka 1 to 10 Divide by
unit to hecto 1 to 100 Divide by 2
unit to kilo 1 to 1,000 Divide by 3
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unit to deci
1 to 0.1 Multiply by
unit to centi 1 to 0.01 Multiply by 2
unit to milli 1 to 0.001 Multiply by 3
Conversion Table
Listed below, in the unit conversion table, are some of the common metric
units of measure.
Unit Conversion Table
1 kilometer (km) = 1,000 meters (m)
1 hectometer (hm) = 100 meters
1 dekameter (dam) = 10 meters
1 meter (m)
Length
1 decimeter (dm) = 10 meter
1 centimeter (cm) = 100 meter
1 millimeter (mm) = 1,000 meter
1 kilogram (kg) = 1,000 grams (g)
1 hectogram (hg) = 100 grams
1 dekagram (dag) = 10 grams
1 gram (g)
Mass
1 decigram (dg) = 10 gram
1 centigram (cg) = 100 gram
1 milligram (mg) = 1,000 gram
1 kiloliter (kL) = 1,000 liters (L)
1 hectoliter (hL) = 100 liters
1 dekaliter (daL) = 10 liters
1 liter (L)
Volume
1 deciliter (dL) = 10 liter
1 centiliter (cL) = 100 liter
1 milliliter (mL) = 1,000 liter
1,000 × 1m
1,000 × 1g
100 × 1g
10 × 1g
1 × 1g
.1 × 1g
.01 × 1g
.001 × 1g
1,000 × 1L
Distinction Between Mass and Weight
There is a distinction between mass and weight. The weight of a body is
related to gravity whereas the mass of a body is not. For example, your
weight on the earth is different than it is on the moon, but your mass is the
same in both places. Mass is a measure of a body's resistance to motion. The
more massive a body, the more resistant it is to motion. Also, more massive
bodies weigh more than less massive bodies.
Converting Metric Units
To convert from one metric unit to another metric unit:
1. Determine the location of the original number on the metric scale
(pictured in each of the following examples).
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Multiply or divide the original number by 10 raised to the same number
of places as is necessary to move to the metric unit you wish to go to. (If
changing from a larger to a smaller unit, multiply by 10. If changing from a
smaller to a larger unit, divide by 10)
We can also convert from one metric unit to another using unit fractions.
Both methods are shown in the following example.
Example 1
Convert 3 kilograms to grams.
Solution
a. 3 kg can be written as 3.0 kg. Then,
Since we are moving from a larger unit to a smaller unit. Multiply the original
number by 10 three times.
grams
Thus, .
3 kg = 3,000 g
b. We can also use unit fractions to make this conversion.
Since we are converting to grams, and . we choose the unit fraction 1,000
g since grams is in the numerator.
1,000 g = 1 kg
1 kg
3 kg 3 kg ⋅1,000 g
1 kg
1,000 g
3 kg ⋅ 1 kg
3 ⋅ 1, 000 g
3, 000 g
Example 2
Convert 67.2 hectoliters to milliliters.
Solution
Since we are moving from a larger unit to a smaller unit. Multiply the original
number by 10 five times.
67.2 × 105 milliliters
= 6, 720, 000
Thus, .
67.2 hL = 6,720,000 mL
Example 3
Convert 100.07 centimeters to meters.
Solution
Since we are moving from a smaller unit to a larger unit. Divide the original
number by 10 two times.
meters
100.07 ÷ 10 = 1.0007
Thus, .
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Example 4
Convert 0.16 milligrams to grams.
Solution
Since we are moving from a smaller unit to a larger unit. Divide the original
number by 10 three times.
grams
0.16 ÷ 10 = 0.00016
Thus,
.
0.16 mg = 0.00016
Example 5
A medication is sold in 5 milligram size capsules. The pharmacist receives
the medication that that is used to fill the capsules in 3.5 kilogram bags. How
many capsules can be produced?
Solution
First we need to convert from kilograms to milligrams. Since we are moving
from a larger to a smaller unit. Multiply the original number by 10 six times.
3.5 × 106 = 3, 500, 000 milligrams
Each capuse holds 5 milligrams, so we divide the number we just found by 5
to determine the number of capsules that can be filled.
capsules
3, 500, 000 ÷ 5 = 700, 000
Thus, 3.5 kilograms of the medication produce 700,000 capsules.
Try It Now 1
Make the following conversions. If a fraction occurs, convert it to a decimal
rounded to two decimal places.
411 kilograms to grams
5.626 liters to centiliters
80 milliliters to kiloliters
150 milligrams to centigrams
2.5 centimeters to meters
Answer
411,000 g
562.6 cL
0.00008 kL
15 cg
0.025 m
This page titled 1.1.2: The Metric System of Measurement is shared under a
CC BY license and was authored, remixed, and/or curated by Leah Griffith,
Veronica Holbrook, Johnny Johnson & Nancy Garcia.
1.1.2.4 146701
1.1.2.1: Exercises
In the metric system, the prefix tells what power of ten the basic unit is
being multiplied by. What power of ten does the prefix mean?
milli
kilo
hecto
centi
giga f. mega
deci
State the symbol or abbreviation for each prefix listed in number 1.
Convert the following:
9 meters to centimeters
520 milliliters to liters
48.66 liters to deciliters
2 millimeters to hectometers
211 centimeters to meters f. 6325 milliliters to liters
1278 grams to kilograms
3.1 kilograms to grams
67 millimeters to centimeters
1.24 kilograms to milligrams
6 kilograms to grams.
99.04 dekaliters to centiliters
A wire costs $2 per meter. How much will 3 kilometers of wire cost?
This page titled 1.1.2.1: Exercises
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1.1.3: Conversion Between the Metric and US Customary Systems of
Measurement
6.1.3 Learning Objectives
To estimate equivalence between the Metric Units and US Customary Units
To convert Metric Units to US Customary Units To convert US Customary
Units to Metric Units
Estimate equivalence between the Metric Units and US Customary Units
Before converting metric units to US customary units and vice versa, you
need to familiarize the approximate metric equivalents of common
customary units. The table below summarizes the commonly used metric
and US customary units and their approximate equivalence. The notations
enclosed are the abbreviations of each unit.
Metric UnitsUS Customary Units
Lengths
1 meter (m) 3.28 feet (ft. or ’)
1 kilometer (km) 0.62 mile (m)
1 centimeter (cm) 0.39 inch (in. or ”)
0.9144 meter (m)1 yard (yd.)
1.852 kilometers (km) 1 nautical mile (nmi. or NM)
Weight
1 kilogram (kg) 2.2 pounds (lb. or #)
28.3 grams (g) 1 ounce (oz.)
1 milligram (mg) 0.002 teaspoon (tsp.)
Fluid Capacity
29.57 milliliters (mL) 1 fluid ounce (fl. Oz.)
1 liter (L) 4.23 cups (c)
3.79 liters (L) 1 gallon (gal.)
473.176 milliliters (mL) 1 pint (pt.)
0.95 liters (L) 1 quart (qt.)
158.987 liters (L) 1 barrel (bbl.)
Additional Common Conversions
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Metric Units
US Customary Units
Area
6.45 square centimeters (cm2) 1 square inch (in2)
1 square meter (m2) 1.196 square yard (yd2)
4046.873 square meters (m2) 1 acre (ac.)
Volume and Capacity
16.39 milliliter (mL) 1 cubic inch (in3)
28.32 liters (L) 1 cubic foot (ft3)
764.55 liters (L) 1 cubic yard (yd3)
1233.482 cubic meters (m3) 1 cubic foot (ft3)
Convert Metric Units to US Customary Units
You have familiarized the commonly used equivalence between metric and
US customary units. At this point, you will learn to convert metric units to US
customary units.
To convert metric units to US customary units, use their estimate
equivalence. Write the equivalence as a ratio, and multiply it to the given
measurement (in metric) to convert. Be sure that the units of measure to
convert cancels out. For instance, to convert liters to gallons, be sure to
cancel out liters and retain gallons as the final unit.
Now, let us consider some examples.
Example 1
The official regulators of a shipment company classify a parcel weighing
more than 51 pounds (lbs.) as a heavy shipment. Andrea packed 24
terracotta plant containers to be shipped to her customers. If each plant
container weighs 1.3 kilograms (kg), will her package be classified as a
heavy shipment?
Solution
To determine whether the parcel is a heavy shipment or not, you need to
convert the total weight of the plant containers to pounds.
Compute for the total weight.
Total Weight = 24 ×1.3 kg
= 31.2 kg
Convert the total weight to pounds. Use the metric to US customary units’
equivalence: 1 kg = 2.2 lbs.
31.2 kg × 1 kg = 68.64 lbs
Since the total weight is more than 51 lbs., the shipment is considered
heavy.
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Try it Now 1
Angelica who lives in Las Vegas, Nevada, visited her friend in Canada. She
knows that Canada uses the metric system, so their speed limits are posted
in kilometers per hour. On her way, she sees a road sign that says, “max 60
km/h”. She knows the sign means “a maximum speed of 60 kilometers per
hour” but she is unsure if this is over the speed limit in Las Vegas, Nevada.
How can Angelica convert 60 km/h to miles per hour? Use the metric to US
customary units’ equivalence: 1 kilometer (km) = 0.62 mile.
Answer
In Las Vegas, the limit is 65 miles per hour (mph). Since 37.2 mph is below
this limit, 60km/h is not over the speed limit.
Example 2
Half of a meter of ribbon is required for wrapping presents enclosed in a
rectangular box. How many presents using the same box size can be
wrapped with 10 yards of ribbon?
Solution
Convert half a meter (0.5 m) to yards. Use the metric to US customary units’
equivalence: 1 yard = 0.9144 meter.
1 yd
0.5 m × 0.9144 = 0.5468 yd
The result means that one present requires 0.5468 yards of ribbon. Calculate
the number of presents by dividing 10 yards by 0.5468 yards.
0.5468 yd 1 present
10 yds ÷ 1 present = 10 yd × 0.5468 yd = 18.29 presents
Approximately, 18 presents can be wrapped with 10 yards of ribbons.
Try it Now 2
Crystal and her friends are planning for a road trip, which they have
determined to be 2,000 kilometers (km) long. They want to rent a van that
can travel 100 km using 9 liters of gas. Suppose the gas price is $2.74 a
gallon (gal.), how many gallons of gas they will need? How much money will
the group of friends need to save for gas? Use the metric to US customary
units’ equivalence: 4.405 liters (L)= 1 gallon (gal.).
