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Multiple Unit Multipliers: English System of Measurement

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions, such as Grand Rapids Community College, Pikes Peak Community College, and Austin Peay State University.

Unit multipliers are a great tool for converting units. In this lesson, we will explore using multiple unit multipliers to convert units within the English system of measurement through definition and examples.

English System of Measurement

Have you ever estimated the length or distance of something using your feet? For instance, measuring the length of a room by counting how many steps it takes to get from one side to the other.

This is actually how the English system of measurement came about. This system of measurement was developed when people used to measure things using their body parts, like their feet, and everyday household items, like a cup.


engunitmult1


The main units of the English measurement system, shown in the image, are related by conversion facts. For example, 12 inches is equal to 1 foot and 16 ounces is equal to 1 pound. Some common conversion facts in the English system of measurement are shown in the image.


engunitmult2


These conversion facts lead to a useful tool called unit multipliers. Let's explore!

Unit Multipliers

Anytime we divide a number or quantity by an equal number or quantity, the result is 1. This concept applies to measurement units as well. Recall that we said that 12 inches is equal to 1 foot. Therefore, if we divide 12 inches by 1 foot or we divide 1 foot by 12 inches, we get 1.

  • 12 in / 1 ft = 1 ft / 12 in = 1

Fractions that equal 1, such as these, are called unit multipliers. These come in extremely handy when we want to convert between units.


engunitmult3


Notice that since unit multipliers are actually equal to 1, we can multiply them by another quantity without changing that quantity. It is because of this fact that we can use unit multipliers to convert units.

Sample Problem

For example, suppose you have three pounds of chicken, but the recipe you are making calls for 30 ounces of chicken. How do you know if you have enough? In this instance, we want to convert 3 pounds to ounces. We have the conversion fact that 16 ounces is equal to 1 pound. The unit multipliers that correspond to this fact are 16 oz / 1 lb and 1 lb / 16 oz. Since we want the pounds to cancel, we should multiply 3 pounds by 16 oz / 1 lb, because the pound unit in the numerator and in the denominator will cancel, leaving ounces.


engunitmult4


Perfect! The pounds units cancel, leaving ounces, so we get that 3 pounds is equal to 48 ounces, so there is definitely enough chicken to make the recipe!

In general, to use unit multipliers to convert units, we use the following steps:

  1. Use conversion facts to find a unit multiplier that involves the unit you are changing from and the unit you are changing to.
  2. Multiply the quantity you are converting by the unit multiplier that will cancel out the unit you are changing from and leave the unit you are changing to.
  3. Simplify.

Sometimes, it takes more than one unit multiplier to convert units. Let's take a look at how to handle these situations.

Multiple Unit Multipliers

Suppose you are at a high school basketball game. The court is marked as being 28 yards long. A friend asks you how long you think that would be in inches. Hmmm…this one isn't so obvious. We want to convert from yards to inches, but we don't have a common conversion fact that relates yards directly to inches. We can get around this by using multiple unit multipliers, or more than one unit multiplier. This is very similar to using just one unit multiplier. We use the following steps:

  1. Use conversion facts to find multiple unit multipliers that involve the unit you are changing from, the unit you are changing to, and any units that may be needed in between to make the conversion.
  2. Multiply the quantity you are converting by all of the unit multipliers so that all of the units that you don't want to end up with cancel out, and you are left with the unit you are changing to.
  3. Simplify.

Consider the basketball court again. Using the conversion facts that relate yards, feet, and inches, we get the following unit multipliers.

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