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9.1: Measurement and the United States System

Last updated

Jun 8, 2024
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- 9: Measurement and Geometry
- 9.2: The Metric System of Measurement

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Learning Objectives

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.

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.

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.

Definition: Unit Fraction

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 $\dfrac{\text{1 wk}}{\text{7 da}}$

Sample Set A

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, $\text{11 yd = 33 ft}$ .

Sample Set A

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, $\text{36 fl oz = 2.25 pt}$ .

Sample Set A

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: $60 (50 - 4) = 3,000 - 240 = 2,760$

Learning Objectives

Measurement

Definition: Measurement

Definition: Standard Unit of Measure

The United States System of Measurement

Conversions in the United States System

Definition: Unit Fraction

Sample Set A

Sample Set A

Sample Set A

Exercises

Exercises for Review

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