Converting U.S. Units Lesson

Measurements allow us to use numerical values to describe an observed quantity. In most parts of the world, the metric system is used for measuring various quantities. In the United States, we use U.S. customary units. In this lesson, we will focus on how to convert between U.S. units of measurement. Let’s look at the most commonly used U.S. units of measurement:

U.S. Units
Weight	Length	Volume	Time
ounces	inches	fluid ounces	seconds
pounds	feet	cups	minutes
tons	yards	pints	hours
	miles	quarts	days
		gallons	weeks

U.S. Units of Weight (Mass)

U.S. Units of Weight (Mass)
16 ounces	=	1 pound
2000 pounds	=	1 ton

U.S. Units of Length

U.S. Units of Length
12 inches	=	1 foot
3 feet	=	1 yard
1760 yards	=	1 mile

U.S. Units of Volume

U.S. Units of Volume
8 fluid ounces	=	1 cup
2 cups	=	1 pint
2 pints	=	1 quart
4 quarts	=	1 gallon

U.S. Units of Time

U.S. Units of Time
60 seconds	=	1 minute
60 minutes	=	1 hour
24 hours	=	1 day
7 days	=	1 week

Most of us have been in a situation in which we needed to change between units of measurement. As an example, suppose we are baking and our recipe calls for 1 pint of butter. Let's suppose, our measurement device is labeled using cups. What can we do in this scenario? We would convert 1 pint into 2 cups so that we can create our baked goods.
In order to perform a unit conversion, we need to understand the relationship between the units involved in our conversion. We can then set up a unit fraction. A unit fraction has a value of 1 and will be used to show the relationship between the two units. As an example, 12 inches is equal to 1 foot. We can set up a unit fraction as either: $$\frac{12 \hspace{.25em}\text{inches}}{1 \hspace{.25em}\text{foot}}\hspace{.25em}\text{or}\hspace{.25em}\frac{1 \hspace{.25em}\text{foot}}{12 \hspace{.25em}\text{inches}}$$ Since the value in the numerator is the same as the value in the denominator, the value of the fraction is 1. Recall that we can multiply any number by one and the result is unchanged. In order to convert between U.S. customary units of measurement:

• Set up the unit fraction with the desired units in the numerator and the unwanted units in the denominator
• Multiply the quantity desired to be converted by the unit fraction
  • The undesired units will cross cancel

Let's take a look at a few examples.
Example 1: Convert 48 inches to feet.
To solve this problem, we can look at our table above for U.S. units of length. We see that 1 foot is equal to 12 inches. Since we desire our unit of measurement to be in feet, we will set up a unit fraction where feet are in the numerator and inches are in the denominator: $$\frac{1 \hspace{.25em}\text{foot}}{12 \hspace{.25em}\text{inches}}$$ Now, we will multiply 48 inches by our unit fraction: $$\require{cancel}48\hspace{.25em}\text{inches}\cdot \frac{1 \hspace{.25em}\text{foot}}{12 \hspace{.25em}\text{inches}}$$ Our undesired units will cancel: $$\cancel{48}4\hspace{.25em}\cancel{\text{inches}}\cdot \frac{1 \hspace{.25em}\text{foot}}{\cancel{12}\hspace{.25em}\cancel{\text{inches}}}$$ Now we can multiply 4 x 1 foot (foot will become feet when plural): $$4 \cdot 1 \hspace{.25em}\text{foot}=4 \hspace{.25em}\text{feet}$$ 48 inches is equal to 4 feet.
Example 2: Convert 14 gallons to pints.
In some cases, we will need to perform two unit conversions in order to get from one unit to another. To solve this problem, we can look at our table above for U.S. units of volume. We see that 1 gallon is equal to 4 quarts and 1 quart is equal to 2 pints. Since we desire our unit of measurement to be in pints, we will need to convert twice, once from gallons to quarts, and then again from quarts to pints. This process can be done as one step, but we will use two for the sake of clarity. Let's begin by setting up a unit fraction to show the relationship between gallons and quarts. Quarts represents the desired measurement, so we place quarts in the numerator and gallons in the denominator: $$\frac{4 \hspace{.25em}\text{quarts}}{1 \hspace{.25em}\text{gallon}}$$ Now we will multiply 14 gallons by our unit fraction: $$14\hspace{.25em}\text{gallons}\cdot \frac{4 \hspace{.25em}\text{quarts}}{1 \hspace{.25em}\text{gallon}}$$ Our undesired units will cancel: $$14\hspace{.25em}\cancel{\text{gallons}}\cdot \frac{4 \hspace{.25em}\text{quarts}}{1 \hspace{.25em}\cancel{\text{gallon}}}$$ Note: don't worry about singular gallon vs plural gallons, the units here are what cancel.
Now we can multiply 14 x 4 quarts: $$14 \cdot 4 \hspace{.25em}\text{quarts}=56 \hspace{.25em}\text{quarts}$$ 14 gallons is equal to 56 quarts.
Now we are ready to take the next step and convert quarts to pints. We know that 2 pints are equal to 1 quart. We will set up our unit fraction with pints our desired units in the numerator, and quarts our undesired units in the denominator: $$\frac{2 \hspace{.25em}\text{pints}}{1 \hspace{.25em}\text{quart}}$$ Now we will multiply 56 quarts by our unit fraction: $$56\hspace{.25em}\text{quarts}\cdot \frac{2 \hspace{.25em}\text{pints}}{1 \hspace{.25em}\text{quart}}$$ Our undesired units will cancel: $$56\hspace{.25em}\cancel{\text{quarts}}\cdot \frac{2 \hspace{.25em}\text{pints}}{1 \hspace{.25em}\cancel{\text{quart}}}$$ Note: don't worry about singular quart vs plural quarts, the units here are what cancel.
Now we can multiply 56 x 2 pints: $$56 \cdot 2 \hspace{.25em}\text{pints}=112 \hspace{.25em}\text{pints}$$ 56 quarts is equal to 112 pints.
In terms of our original problem:
14 gallons is equal to 112 pints.
Example 3: Convert 160,000 ounces to tons.
To solve this problem, we can look at our table above for U.S. units of weight (mass). We see that 1 ton is equal to 2000 pounds and 1 pound is equal to 16 ounces. We need to go from ounces to pounds and then from pounds to tons. This problem can be done in one step by multiplying by two unit fractions. This means setting up the following two unit fractions: $$\frac{1 \hspace{.25em}\text{lb}}{16 \hspace{.25em}\text{oz}}, \frac{1 \hspace{.25em}\text{ton}}{2000 \hspace{.25em}\text{lb}}$$ Note: ounce (oz), pound (lb)
Now we will multiply 160,000 ounces by our two unit fractions: $$160,\hspace{-.2em}000 \hspace{.25em}\text{oz}\cdot \frac{1 \hspace{.25em}\text{lb}}{16 \hspace{.25em}\text{oz}}\cdot \frac{1 \hspace{.25em}\text{ton}}{2000 \hspace{.25em}\text{lb}}$$ Our undesired units will cancel: $$160,\hspace{-.2em}000 \hspace{.25em}\cancel{\text{oz}}\cdot \frac{1 \hspace{.25em}\cancel{\text{lb}}}{16 \hspace{.25em}\cancel{\text{oz}}}\cdot \frac{1 \hspace{.25em}\text{ton}}{2000 \hspace{.25em}\cancel{\text{lb}}}$$ Now our simplified problem becomes: $$\frac{\cancel{160,\hspace{-.1em}000}5 \hspace{.25em}\text{tons}}{\cancel{32,\hspace{-.1em}000}}=5\hspace{.25em}\text{tons}$$ 160,000 ounces is equal to 5 tons.

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