Lesson Objectives

- Demonstrate an understanding of fractions
- Learn how to set up a unit fraction
- Learn how to convert between U.S. customary units of measurement

## How to Convert between U.S. Customary Units of Measurement

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:

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: $$\frac{12 \hspace{.25em}inches}{1 \hspace{.25em}foot}\hspace{.25em}or\hspace{.25em}\frac{1 \hspace{.25em}foot}{12 \hspace{.25em}inches}$$ 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:

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: $$\frac{1 \hspace{.25em}foot}{12 \hspace{.25em}inches}$$ Now we will multiply 48 inches by our unit fraction: $$\require{cancel}48\hspace{.25em}inches \cdot \frac{1 \hspace{.25em}foot}{12 \hspace{.25em}inches}$$ Our undesired units will cancel: $$\cancel{48}4\hspace{.25em}\cancel{inches}\cdot \frac{1 \hspace{.25em}foot}{\cancel{12}\hspace{.25em}\cancel{inches}}$$ Now we can multiply 4 x 1 foot (foot will become feet when plural): $$4 \cdot 1 \hspace{.25em}foot=4 \hspace{.25em}feet$$ 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: $$\frac{4 \hspace{.25em}quarts}{1 \hspace{.25em}gallon}$$ Now we will multiply 14 gallons by our unit fraction: $$14\hspace{.25em}gallons \cdot \frac{4 \hspace{.25em}quarts}{1 \hspace{.25em}gallon}$$ Our undesired units will cancel: $$14\hspace{.25em}\cancel{gallons}\cdot \frac{4 \hspace{.25em}quarts}{1 \hspace{.25em}\cancel{gallon}}$$ Note: don't worry about singular gallon vs plural gallons, the units here are what cancel.

Now we can multiply 14 x 4 quarts: $$14 \cdot 4 \hspace{.25em}quarts=56 \hspace{.25em}quarts$$ 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: $$\frac{2 \hspace{.25em}pints}{1 \hspace{.25em}quart}$$ Now we will multiply 56 quarts by our unit fraction: $$56\hspace{.25em}quarts \cdot \frac{2 \hspace{.25em}pints}{1 \hspace{.25em}quart}$$ Our undesired units will cancel: $$56\hspace{.25em}\cancel{quarts}\cdot \frac{2 \hspace{.25em}pints}{1 \hspace{.25em}\cancel{quart}}$$ Note: don't worry about singular quart vs plural quarts, the units here are what cancel.

Now we can multiply 56 x 2 pints: $$56 \cdot 2 \hspace{.25em}pints=112 \hspace{.25em}pints$$ 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: $$\frac{1 \hspace{.25em}lb}{16 \hspace{.25em}oz}, \frac{1 \hspace{.25em}ton}{2000 \hspace{.25em}lb}$$ Note: ounce (oz), pound (lb)

Now we will multiply 160,000 ounces by our two unit fractions: $$160,\hspace{-.2em}000 \hspace{.25em}oz \cdot \frac{1 \hspace{.25em}lb}{16 \hspace{.25em}oz}\cdot \frac{1 \hspace{.25em}ton}{2000 \hspace{.25em}lb}$$ Our undesired units will cancel: $$160,\hspace{-.2em}000 \hspace{.25em}\cancel{oz}\cdot \frac{1 \hspace{.25em}\cancel{lb}}{16 \hspace{.25em}\cancel{oz}}\cdot \frac{1 \hspace{.25em}ton}{2000 \hspace{.25em}\cancel{lb}}$$ Now our simplified problem becomes: $$\frac{\cancel{160,\hspace{-.1em}000}5 \hspace{.25em}tons}{\cancel{32,\hspace{-.1em}000}}=5\hspace{.25em}tons$$ 160,000 ounces is equal to 5 tons.

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 |

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: $$\frac{12 \hspace{.25em}inches}{1 \hspace{.25em}foot}\hspace{.25em}or\hspace{.25em}\frac{1 \hspace{.25em}foot}{12 \hspace{.25em}inches}$$ 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

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: $$\frac{1 \hspace{.25em}foot}{12 \hspace{.25em}inches}$$ Now we will multiply 48 inches by our unit fraction: $$\require{cancel}48\hspace{.25em}inches \cdot \frac{1 \hspace{.25em}foot}{12 \hspace{.25em}inches}$$ Our undesired units will cancel: $$\cancel{48}4\hspace{.25em}\cancel{inches}\cdot \frac{1 \hspace{.25em}foot}{\cancel{12}\hspace{.25em}\cancel{inches}}$$ Now we can multiply 4 x 1 foot (foot will become feet when plural): $$4 \cdot 1 \hspace{.25em}foot=4 \hspace{.25em}feet$$ 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: $$\frac{4 \hspace{.25em}quarts}{1 \hspace{.25em}gallon}$$ Now we will multiply 14 gallons by our unit fraction: $$14\hspace{.25em}gallons \cdot \frac{4 \hspace{.25em}quarts}{1 \hspace{.25em}gallon}$$ Our undesired units will cancel: $$14\hspace{.25em}\cancel{gallons}\cdot \frac{4 \hspace{.25em}quarts}{1 \hspace{.25em}\cancel{gallon}}$$ Note: don't worry about singular gallon vs plural gallons, the units here are what cancel.

Now we can multiply 14 x 4 quarts: $$14 \cdot 4 \hspace{.25em}quarts=56 \hspace{.25em}quarts$$ 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: $$\frac{2 \hspace{.25em}pints}{1 \hspace{.25em}quart}$$ Now we will multiply 56 quarts by our unit fraction: $$56\hspace{.25em}quarts \cdot \frac{2 \hspace{.25em}pints}{1 \hspace{.25em}quart}$$ Our undesired units will cancel: $$56\hspace{.25em}\cancel{quarts}\cdot \frac{2 \hspace{.25em}pints}{1 \hspace{.25em}\cancel{quart}}$$ Note: don't worry about singular quart vs plural quarts, the units here are what cancel.

Now we can multiply 56 x 2 pints: $$56 \cdot 2 \hspace{.25em}pints=112 \hspace{.25em}pints$$ 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: $$\frac{1 \hspace{.25em}lb}{16 \hspace{.25em}oz}, \frac{1 \hspace{.25em}ton}{2000 \hspace{.25em}lb}$$ Note: ounce (oz), pound (lb)

Now we will multiply 160,000 ounces by our two unit fractions: $$160,\hspace{-.2em}000 \hspace{.25em}oz \cdot \frac{1 \hspace{.25em}lb}{16 \hspace{.25em}oz}\cdot \frac{1 \hspace{.25em}ton}{2000 \hspace{.25em}lb}$$ Our undesired units will cancel: $$160,\hspace{-.2em}000 \hspace{.25em}\cancel{oz}\cdot \frac{1 \hspace{.25em}\cancel{lb}}{16 \hspace{.25em}\cancel{oz}}\cdot \frac{1 \hspace{.25em}ton}{2000 \hspace{.25em}\cancel{lb}}$$ Now our simplified problem becomes: $$\frac{\cancel{160,\hspace{-.1em}000}5 \hspace{.25em}tons}{\cancel{32,\hspace{-.1em}000}}=5\hspace{.25em}tons$$ 160,000 ounces is equal to 5 tons.

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