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Measurement and Geometry: 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.

Table 1
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.

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,

Table 2
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.

Solution

6 yd

Exercise 2

Convert 2 mi to feet.

Solution

10,560 ft

Exercise 3

Convert 26 ft to yards.

Solution

8.67 yd

Exercise 4

Convert 9 qt to pints.

Solution

18 pt

Exercise 5

Convert 52 min to hours.

Solution

0.87 hr

Exercise 6

Convert 412 hr to weeks.

Solution

2.45 wk

Exercises

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

Exercise 7

14 yd to feet

Solution

42 feet

Exercise 8

3 mi to yards

Exercise 9

8 mi to inches

Solution

506,880 inches

Exercise 10

2 mi to inches

Exercise 11

18 in. to feet

Solution

1.5 feet

Exercise 12

84 in. to yards

Exercise 13

5 in. to yards

Solution

0.14 yard

Exercise 14

106 ft to miles

Exercise 15

62 in. to miles

Solution

0.00 miles (to two decimal places)

Exercise 16

0.4 in. to yards

Exercise 17

3 qt to pints

Solution

6 pints

Exercise 18

5 lb to ounces

Exercise 19

6 T to ounces

Solution

192,000 ounces

Exercise 20

4 oz to pounds

Exercise 21

15,000 oz to pounds

Solution

937.5 pounds

Exercise 22

15,000 oz to tons

Exercise 23

9 tbsp to teaspoons

Solution

27 teaspoons

Exercise 24

3 c to tablespoons

Exercise 25

5 pt to fluid ounces

Solution

80 fluid ounces

Exercise 26

16 tsp to cups

Exercise 27

5 fl oz to quarts

Solution

0.16 quart

Exercise 28

3 qt to gallons

Exercise 29

5 pt to teaspoons

Solution

480 teaspoons

Exercise 30

3 qt to tablespoons

Exercise 31

18 min to seconds

Solution

1,080 seconds

Exercise 32

4 days to hours

Exercise 33

3 hr to days

Solution

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

Exercise 34

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.

Solution

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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