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6.1.1: Measurement and the United States System

  • Page ID
    87298
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    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:

    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.

    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 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 Capacity 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

    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 \(\dfrac{\text{1 ft}}{\text{12 in}}\) 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 \(\dfrac{\text{1 wk}}{\text{7 da}}\)

    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 \(\text{1 yd = 3 ft}\). 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.

    \(\begin{array} {rcll} {\text{11 yd}} & = & {\dfrac{\text{11 yd}}{1} \cdot \dfrac{\text{3 ft}}{\text{1 yd}}} & {\text{Divide out common units.}} \\ {} & = & {\dfrac{\text{11 } \cancel{\text{yd}}}{1} \cdot \dfrac{\text{3 ft}}{\text{1 } \cancel{\text{yd}}}} & {\text{(Units can be added, subtracted, multiplied, and divided, just as numbers can.)}} \\ {} & = & {\dfrac{11 \cdot 3 \text{ ft}}{1}} & {} \\ {} & = & {\text{33 ft}} & {} \end{array}\)

    Thus, \(\text{11 yd = 33 ft}\).

    Example 2

    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.

    \(\begin{array} {rcll} {\text{36 fl oz}} & = & {\dfrac{\text{36 fl oz}}{1} \cdot \dfrac{1 \text{ pt}}{\text{16 fl oz}}} & {\text{Divide out common units.}} \\ {} & = & {\dfrac{36 \cancel{\text{ fl oz}}}{1} \cdot \dfrac{\text{1 pt}}{16 \cancel{\text{ fl oz}}}} & {} \\ {} & = & {\dfrac{36 \cdot 1 \text{ pt}}{16}} & {} \\ {} & = & {\dfrac{\text{36 pt}}{16}} & {\text{Reduce}} \\ {} & = & {\dfrac{9}{4} \text{ pt}} & {\text{Convert to decimals: } \dfrac{9}{4} = 2.25} \end{array}\)

    Thus, \(\text{36 fl oz = 2.25 pt}\).

    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 \(\dfrac{1 \text{ yd}}{3 \text{ ft}}\).

    \(\begin{array} {rcll} {\text{18 ft}} & = & {\dfrac{\text{18 ft}}{1} \cdot \dfrac{1 \text{ yd}}{\text{3 ft}}} & {\text{Divide out common units.}} \\ {} & = & {\dfrac{18 \cancel{\text{ ft}}}{1} \cdot \dfrac{\text{1 yd}}{\text{3 } \cancel{\text{ft}}}} & {} \\ {} & = & {\dfrac{18 \text{ yd}}{1\cdot 3}} & {} \\ {} & = & {\text{6 yd}} & {} \end{array}\)

    Thus, \(\text{18 ft= 6 yd}\). You would need to purchase 6 yards of fabric for the red carpet.

    Try It Now 1

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

    1. Convert 2 mi to feet.
    2. Convert 26 ft to yards.
    3. Convert 9 qt to pints.
    Answer
    1. 10,560 ft
    2. 8.67 yd.
    3. 18 pt

    This page titled 6.1.1: 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.