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# Measurement and the United States System

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses the United States System of measurement. By the end of the module students should know what the word measurement means, be familiar with United States system of measurement and be able to convert from one unit of measure in the United States system to another unit of measure.

## Section Overview

• Measurement
• The United States System of Measurement
• Conversions in the United States System

## 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

Measurement is comparison to some standard.

### 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:

1. Accessibility. We should have access to the standard so we can make comparisons.
2. Invariance. We should be confident that the standard is not subject to change.
3. 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 cups1 quart (qt) = 2 pints1 gallon (gal) = 4 quarts Time 1 minute (min) = 60 seconds (sec) 1 hour ( hr) = 60 minutes1 day (da) = 24 hours1 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.

### 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 1ft = 12in. 1ft = 12in. size 12{"1ft"="12in"} {} 1ft 12in. or 12in. 1ft 1ft 12in. or 12in. 1ft size 12{ { {"1ft"} over {"12in."} } "or" { {"12in."} over {"1ft"} } } {} 1pt = 16 fl oz 1pt = 16 fl oz size 12{"1pt"="16 fl oz"} {} 1pt 16 fl oz or 16 fl oz 1pt 1pt 16 fl oz or 16 fl oz 1pt size 12{ { {"1pt"} over {"16 fl oz"} } " or " { {"16 fl oz"} over {"1pt"} } } {} 1wk = 7da 1wk = 7da size 12{"1wk"="7da"} {} 7da 1wk or 1wk 7da 7da 1wk or 1wk 7da size 12{ { {"7da"} over {"1wk"} } " or " { {"1wk"} over {"7da"} } } {}

### Sample Set A

Make the following conversions. If a fraction occurs, convert it to a decimal rounded to two decimal places.

#### Example 1

Convert 11 yards to feet.

Looking in the unit conversion table under length, we see that 1 yd =3 ft1 yd =3 ft size 12{1" yd "="3 ft"} {}. There are two corresponding unit fractions, 1 yd3 ft1 yd3 ft size 12{ { {"1 yd"} over {"3 ft"} } } {}and 3 ft1 yd3 ft1 yd size 12{ { {"3 ft"} over {"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 3 ft1 yd3 ft1 yd size 12{ { {"3 ft"} over {"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.

11 yd = 11yd13ft1yd Divide out common units. =11yd13ft1yd (Units can be added, subtracted, multiplied, and divided, just as numbers can.) =113ft1 =33ft 11 yd = 11yd13ft1yd Divide out common units. =11yd13ft1yd (Units can be added, subtracted, multiplied, and divided, just as numbers can.) =113ft1 =33ft

Thus, 11 yd =33ft11 yd =33ft.

#### Example 2

Convert 36 fl oz to pints.

Looking in the unit conversion table under liquid volume, we see that 1 pt = 16 fl oz1 pt = 16 fl oz size 12{"1 pt "=" 16 fl oz"} {}. Since we are to convert to pints, we will construct a unit fraction with pints in the numerator.

36fl oz = 36 fl oz 1 1 pt 16 fl oz Divide out common units. = 36 fl oz 1 1 pt 16 fl oz = 36 1 pt 16 = 36 pt 16 Reduce. = 9 4 pt Convert to decimals: 9 4 = 2.25 . 36fl oz = 36 fl oz 1 1 pt 16 fl oz Divide out common units. = 36 fl oz 1 1 pt 16 fl oz = 36 1 pt 16 = 36 pt 16 Reduce. = 9 4 pt Convert to decimals: 9 4 = 2.25 .

Thus, 36 fl oz = 2.25 pt36 fl oz = 2.25 pt size 12{"36 fl oz "=" 2" "." "25 pt"} {}.

#### Example 3

Convert 2,016 hr to weeks.

Looking in the unit conversion table under time, we see that 1wk =7da1wk =7da size 12{"1wk "="7da"} {} and that 1 da = 24 hr1 da = 24 hr size 12{1" da "=" 24 hr"} {}. 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 1 da24 hr1 da24 hr size 12{ { {"1 da"} over {"24 hr"} } } {}. The unit fraction needed for converting from days to weeks is 1 wk7 da1 wk7 da size 12{ { {"1 wk"} over {"7 da"} } } {}.

2,016 hr = 2,016 hr 1 1 da 24 hr 1 wk 7 da Divide out common units. = 2,016 hr 1 1 da 24 hr 1 wk 7 da = 2,016 1 wk 24 7 Reduce. = 12 wk 2,016 hr = 2,016 hr 1 1 da 24 hr 1 wk 7 da Divide out common units. = 2,016 hr 1 1 da 24 hr 1 wk 7 da = 2,016 1 wk 24 7 Reduce. = 12 wk

Thus, 2,016 hr = 12 wk2,016 hr = 12 wk size 12{"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.

#### Exercise 1

Convert 18 ft to yards.

6 yd

#### Exercise 2

Convert 2 mi to feet.

10,560 ft

#### Exercise 3

Convert 26 ft to yards.

8.67 yd

#### Exercise 4

Convert 9 qt to pints.

18 pt

#### Exercise 5

Convert 52 min to hours.

0.87 hr

#### Exercise 6

Convert 412 hr to weeks.

2.45 wk

## Exercises

Make each conversion using unit fractions. If fractions occur, convert them to decimals rounded to two decimal places.

14 yd to feet

42 feet

3 mi to yards

8 mi to inches

506,880 inches

2 mi to inches

18 in. to feet

1.5 feet

84 in. to yards

5 in. to yards

0.14 yard

106 ft to miles

### Exercise 15

62 in. to miles

#### Solution

0.00 miles (to two decimal places)

0.4 in. to yards

3 qt to pints

6 pints

5 lb to ounces

6 T to ounces

192,000 ounces

4 oz to pounds

### Exercise 21

15,000 oz to pounds

937.5 pounds

### Exercise 22

15,000 oz to tons

### Exercise 23

9 tbsp to teaspoons

27 teaspoons

### Exercise 24

3 c to tablespoons

### Exercise 25

5 pt to fluid ounces

80 fluid ounces

16 tsp to cups

### Exercise 27

5 fl oz to quarts

0.16 quart

3 qt to gallons

### Exercise 29

5 pt to teaspoons

480 teaspoons

### Exercise 30

3 qt to tablespoons

### Exercise 31

18 min to seconds

1,080 seconds

4 days to hours

### Exercise 33

3 hr to days

#### Solution

18=0.12518=0.125 size 12{ { {1} over {8} } =0 "." "125"} {} day

1212 hr to days

### Exercise 35

1212 da to weeks

#### Solution

114=0.0714114=0.0714 size 12{ { {1} over {"14"} } =0 "." "0714"} {} week

### Exercise 36

3 17 317 wk to seconds

### Exercises for Review

#### Exercise 37

((Reference)) Specify the digits by which 23,840 is divisible.

1,2,4,5,8

#### Exercise 38

((Reference)) Find 245245 size 12{2 { {4} over {5} } } {} of 556556 size 12{5 { {5} over {6} } } {} of 757757 size 12{7 { {5} over {7} } } {}.

#### Exercise 39

((Reference)) Convert 0.3230.323 size 12{0 "." 3 { {2} over {3} } } {} to a fraction.

##### Solution

11301130 size 12{ { {"11"} over {"30"} } } {}

#### Exercise 40

((Reference)) Use the clustering method to estimate the sum: 53+82+79+4953+82+79+49 size 12{"53"+"82"+"79"+"49"} {}.

#### Exercise 41

((Reference)) Use the distributive property to compute the product: 60466046 size 12{"60" cdot "46"} {}.

##### Solution

60(504)=3,000240=2,76060(504)=3,000240=2,760 size 12{"60" $$"50" - 4$$ =3,"000" - "240"=2,"760"} {}

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