9.1: Measurement and the United States System
- know what the word measurement means
- be familiar with United States system of measurement
- be able to convert from one unit of measure in the United States system to another unit of measure
Measurement
There are two major systems of measurement in use today. They are the United States system and the metric system . Before we describe these systems, let's gain a clear understanding of the concept of measurement.
Measurement is comparison to some standard.
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.
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 |
1 foot (ft) = 12 inches (in.)
1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5,280 feet |
| Weight |
1 pound (lb) =16 ounces (oz)
1 ton (T) = 2,000 pounds |
| Liquid Volume |
1 tablespoon (tbsp) = 3 teaspoons (tsp)
1 fluid ounce (fl oz) = 2 tablespoons 1 cup (c) = 8 fluid ounces 1 pint (pt) = 2 cups 1 quart (qt) = 2 pints 1 gallon (gal) = 4 quarts |
| Time |
1 minute (min) = 60 seconds (sec)
1 hour ( hr) = 60 minutes 1 day (da) = 24 hours 1 week (wk) = 7 days |
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 .
A unit fraction is a fraction with a value of 1.
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. | \(\dfrac{\text{1 ft}}{\text{12 in}}\) or \(\dfrac{\text{12 in}}{\text{1 ft}}\) |
| 1 pt = 16 fl oz | \(\dfrac{\text{1 pt}}{\text{16 fl oz}}\) or \(\dfrac{\text{16 fl oz}}{\text{1 pt}}\) |
| 1 wk = 7 da | \(\dfrac{\text{7 da}}{\text{1 wk}}\) or \(\dfrac{\text{1 wk}}{\text{7 da}}\) |
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 \(\text{1 yd = 3 ft}\). There are two corresponding unit fractions, \(\dfrac{\text{1 yd}}{\text{3 ft}}\) and \(\dfrac{\text{3 ft}}{\text{1 yd}}\). Which one should we use? Look to see which unit we wish to convert to. Choose the unit fraction with this unit in the numerator . We will choose \(\dfrac{\text{3 ft}}{\text{1 yd}}\) since this unit fraction has feet in the numerator. Now, multiply 11 yd by the unit fraction. Notice that since the unit fraction has the value of 1, multiplying by it does not change the value of 11 yd.
\(\begin{array} {rcll} {\text{11 yd}} & = & {\dfrac{\text{11 yd}}{1} \cdot \dfrac{30}{\text{1 yd}}} & {\text{Divide out common units.}} \\ {} & = & {\dfrac{\text{11 } \cancel{\text{yd}}}{1} \cdot \dfrac{\text{3 ft}}{\text{1 } \cancel{\text{yd}}}} & {\text{(Units can be added, subtracted, multiplied, and divided, just as numbers can.)}} \\ {} & = & {\dfrac{11 \cdot 3 \text{ ft}}{1}} & {} \\ {} & = & {\text{33 ft}} & {} \end{array}\)
Thus, \(\text{11 yd = 33 ft}\).
Convert 36 fl oz to pints.
Solution
Looking in the unit conversion table under liquid volume , we see that \(\text{1 pt = 16 fl oz}\). Since we are to convert to pints, we will construct a unit fraction with pints in the numerator.
\(\begin{array} {rcll} {\text{36 fl oz}} & = & {\dfrac{\text{36 fl oz}}{1} \cdot \dfrac{1 pt}{\text{16 fl oz}}} & {\text{Divide out common units.}} \\ {} & = & {\dfrac{\text{36 } \cancel{\text{fl oz}}}{1} \cdot \dfrac{\text{1 pt}}{\text{16 } \cancel{\text{fl oz}}}} & {} \\ {} & = & {\dfrac{36 \cdot 1 \text{ pt}}{16}} & {} \\ {} & = & {\dfrac{\text{36 pt}}{16}} & {\text{Reduce}} \\ {} & = & {\dfrac{9}{4} \text{ pt}} & {\text{Convert to decimals: } \dfrac{9}{4} = 2.25} \end{array}\)
Thus, \(\text{36 fl oz = 2.25 pt}\).
Convert 2,016 hr to weeks.
Solution
Looking in the unit conversion table under time , we see that \(\text{1 wk = 7 da}\) and that \(\text{1 da = 24hr}\). To convert from hours to weeks, we must first convert from hours to days and then from days to weeks. We need two unit fractions.
The unit fraction needed for converting from hours to days is \(\dfrac{1 da}{24 hr}\). The unit fraction needed for converting from days to weeks is \(\dfrac{\text{1 wk}}{\text{7 da}}\).
\(\begin{array} {rcll} {\text{2,016 hr}} & = & {\dfrac{\text{2,016 hr}}{1} \cdot \dfrac{\text{1 da}}{\text{24 hr}} \cdot \dfrac{\text{1 wk}}{\text{7 da}}} & {\text{Divide out common units.}} \\ {} & = & {\dfrac{\text{2,016 } \cancel{\text{hr}}}{1} \cdot \dfrac{\text{1 } \cancel{\text{da}}}{\text{24 } \cancel{\text{hr}}} \cdot \dfrac{\text{1 wk}}{\text{7 } \cancel{\text{da}}}} & {} \\ {} & = & {\dfrac{2,016 \cdot 1 \text{ wk}}{24 \cdot 7}} & {\text{Reduce}} \\ {} & = & {\text{12 wk}} & {} \end{array}\)
Thus, \(\text{2,016 hr = 12 wk}\).
Practice Set A
Make the following conversions. If a fraction occurs, convert it to a decimal rounded to two decimal places.
Convert 18 ft to yards.
- Answer
-
6 yd
Practice Set A
Convert 2 mi to feet.
- Answer
-
10,560 ft
Practice Set A
Convert 26 ft to yards.
- Answer
-
8.67 yd
Practice Set A
Convert 9 qt to pints.
- Answer
-
18 pt
Practice Set A
Convert 52 min to hours.
- Answer
-
0.87 hr
Practice Set A
Convert 412 hr to weeks.
- Answer
-
2.45 wk
Exercises
Make each conversion using unit fractions. If fractions occur, convert them to decimals rounded to two decimal places.
Exercise \(\PageIndex{1}\)
14 yd to feet
- Answer
-
42 feet
Exercise \(\PageIndex{2}\)
3 mi to yards
Exercise \(\PageIndex{3}\)
8 mi to inches
- Answer
-
506,880 inches
Exercise \(\PageIndex{4}\)
2 mi to inches
Exercise \(\PageIndex{5}\)
18 in. to feet
- Answer
-
1.5 feet
Exercise \(\PageIndex{6}\)
84 in. to yards
Exercise \(\PageIndex{7}\)
5 in. to yards
- Answer
-
0.14 yard
Exercise \(\PageIndex{8}\)
106 ft to miles
Exercise \(\PageIndex{9}\)
62 in. to miles
- Answer
-
0.00 miles (to two decimal places)
Exercise \(\PageIndex{10}\)
0.4 in. to yards
Exercise \(\PageIndex{11}\)
3 qt to pints
- Answer
-
6 pints
Exercise \(\PageIndex{12}\)
5 lb to ounces
Exercise \(\PageIndex{13}\)
6 T to ounces
- Answer
-
192,000 ounces
Exercise \(\PageIndex{14}\)
4 oz to pounds
Exercise \(\PageIndex{15}\)
15,000 oz to pounds
- Answer
-
937.5 pounds
Exercise \(\PageIndex{16}\)
15,000 oz to tons
Exercise \(\PageIndex{17}\)
9 tbsp to teaspoons
- Answer
-
27 teaspoons
Exercise \(\PageIndex{18}\)
3 c to tablespoons
Exercise \(\PageIndex{19}\)
5 pt to fluid ounces
- Answer
-
80 fluid ounces
Exercise \(\PageIndex{20}\)
16 tsp to cups
Exercise \(\PageIndex{21}\)
5 fl oz to quarts
- Answer
-
0.16 quart
Exercise \(\PageIndex{22}\)
3 qt to gallons
Exercise \(\PageIndex{23}\)
5 pt to teaspoons
- Answer
-
480 teaspoons
Exercise \(\PageIndex{24}\)
3 qt to tablespoons
Exercise \(\PageIndex{25}\)
18 min to seconds
- Answer
-
1,080 seconds
Exercise \(\PageIndex{26}\)
4 days to hours
Exercise \(\PageIndex{27}\)
3 hr to days
- Answer
-
\(\dfrac{1}{8} = 0.125 \text{ day}\)
Exercise \(\PageIndex{28}\)
\(\dfrac{1}{2}\) hr to days
Exercise \(\PageIndex{29}\)
\(\dfrac{1}{2}\) da to weeks
- Answer
-
\(\dfrac{1}{14} = 0.0714 \text{ week}\)
Exercise \(\PageIndex{30}\)
\(3 \dfrac{1}{7}\) wk to seconds
Exercises for Review
Exercise \(\PageIndex{31}\)
Specify the digits by which 23,840 is divisible.
- Answer
-
1,2,4,5,8
Exercise \(\PageIndex{32}\)
Find \(2\dfrac{4}{5}\) of \(5 \dfrac{5}{6}\) of \(7 \dfrac{5}{7}\)
Exercise \(\PageIndex{33}\)
Convert \(0.3 \dfrac{2}{3}\) to a fraction.
- Answer
-
\(\dfrac{11}{30}\)
Exercise \(\PageIndex{34}\)
Use the clustering method to estimate the sum: \(53 + 82 + 79 + 49\).
Exercise \(\PageIndex{35}\)
Use the distributive property to compute the product: \(60 \cdot 46\).
- Answer
-
\(60 (50 - 4) = 3,000 - 240 = 2,760\)