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# 4.6: Customary Units of Measure

Difficulty Level: At Grade Created by: CK-12

## Introduction

Everest in Miles

“You keep telling me that Mount Everest is 29,035 feet high, but how many miles is that?” Teiyisha asked Josh during geography class.

“Well, you don’t need to know the miles because you are climbing it. Doesn’t it make more sense to measure it in feet?” Josh asked while working on his map.

“It might, but if I wanted to know how many miles, how could I figure this out?” Teiyisha asked.

Josh thought about it for a minute.

“You would have to do a conversion. You would have to change feet to miles, and we can do that if we know how many feet are in 1 mile. It still makes more sense to use feet though. Think about it, you can’t exactly draw a straight line up Mount Everest. Miles would be very hard to measure.”

“I get that, but I still want to know how many miles high it is,” Teiyisha explained.

“Alright, first let’s think about miles and feet. There are 5,280 feet in 1 mile.”

Let’s stop there. To accomplish Teiyisha’s task, we will need to create a proportion and convert feet to miles. We can do this by converting among customary units of measurement. Pay attention to this lesson and you will have it figured out by the end.

What You Will Learn

In this lesson you will learn to complete the following skills.

• Convert among customary units of measure.
• Revise or interpret scale drawings, maps, recipes, and models involving the conversion of customary units of measure.
• Solve real-world problems involving scaling of customary units of measure.

Teaching Time

I. Convert among Customary Units of Measure

When we measure in the United States, we often use the customary system of measurement. The customary system is made up of units such as inches, feet, cups, gallons and pounds. You may have also heard of the Metric System of measurement. The metric system is often used in countries outside of the United States or in science measurements. You will learn more about the metric system in another lesson. In this lesson you will learn how to convert between different customary units of measurement. But first, let’s look at some of the units that you are probably already familiar with.

Customary Units of Measurement

Take a few minutes to copy all of these units of measurement down in your notebook.

Now let’s look at how we convert among customary units of measurement. While you may be able to complete some of the mathematics in your head, it may make more sense to use a proportion. Because there is a relationship between different units of measure, you can use proportions to help you convert between customary units of measurement.

First, set up the proportion in the same way you used to find actual measurements from scale drawings. Use the conversion factor as the first ratio, and the known and unknown units in the second ratio.

Example

How many feet are in 5 yards?

Set up a proportion.

The conversion factor is the number of feet in 1 yard: 3 feet1 yard\begin{align*}\frac{3 \ feet}{1 \ yard}\end{align*}.

Now write the second ratio.

The known unit is 5 yards. The unknown unit is x\begin{align*}x\end{align*} feet. Make sure that the second ratio follows the form of the first ratio: feet over yards.

3 feet1 yard=x feet5 yards\begin{align*}\frac{3 \ feet}{1 \ yard}=\frac{x \ feet}{5 \ yards}\end{align*}

Next cross-multiply to solve for x\begin{align*}x\end{align*}.

(1)xx=3(5)=15\begin{align*}(1)x &= 3(5)\\ x &= 15\end{align*}

There are 15 feet in 5 yards.

Example

How many ounces are there is 20 pounds?

First, set up a proportion.

The scale of measurement is 1 pound16 ounces\begin{align*}\frac{1 \ pound}{16 \ ounces}\end{align*}.

The proportion is: 116=20x\begin{align*}\frac{1}{16}= \frac{20}{x}\end{align*}

Next, we cross multiply and solve for the number of ounces.

There are 320 ounces in 20 pounds.

Now that you understand how to use a proportion to complete measurement conversions, we can apply this information to scale drawings, maps, recipes and models.

II. Revise or Interpret Scale Drawings, Maps, Recipes and Models Involving Conversion of Customary Units of Measure

Once you understand that we are going to be using a proportion to solve for missing measurements, you can apply this information when looking at different types of problems. Let’s look at a few examples.

Let’s look at how we can apply our work with proportions when looking at a recipe.

Example

Evan is making a recipe for fruit punch that uses 3 cups of pineapple juice. If he makes 5 batches of the recipe, how many quarts of pineapple juice will he need?

First find the total number of cups he needs.

If there are 3 cups in one batch, and he is making 5 batches, then he will need 3×5=15 cups\begin{align*}3 \times 5 = 15 \ cups\end{align*}.

Set up a proportion.

The conversion factor is the number of cups in a quart: 4 cups1 quart\begin{align*}\frac{4 \ cups}{1 \ quart}\end{align*}.

Now write the second ratio, making sure it follows the form of the first ratio.

4 cups1 quart=15 cupsx quarts\begin{align*}\frac{4 \ cups}{1 \ quart} = \frac{15 \ cups}{x \ quarts}\end{align*}

Next cross-multiply and solve for x\begin{align*}x\end{align*}.

(4)x4xx=1(15)=15=154=334\begin{align*}(4)x &= 1(15)\\ 4x &= 15\\ x &= \frac{15}{4} = 3 \frac{3}{4}\end{align*}

He will need 334\begin{align*}3 \frac{3}{4}\end{align*} quarts of pineapple juice.

We can also use this same method when working with a map. Look at the following example.

Example

The distance from John’s house to Mike’s house on a map is 4.5 inches. The scale of the map is 1.5 inches = 2 miles. What is the actual distance from John’s house to Mike’s house in feet?

First, find the actual distance in miles. Then convert miles to feet.

Write a proportion to find the actual distance between the two houses.

1.5 inches2 miles=4.5 inchesx miles\begin{align*}\frac{1.5 \ inches}{2 \ miles} = \frac{4.5 \ inches}{x \ miles}\end{align*}

Now cross-multiply and solve for x\begin{align*}x\end{align*}.

(1.5)x1.5xx=4.5(2)=9=6\begin{align*}(1.5)x &= 4.5(2)\\ 1.5x &= 9\\ x &= 6\end{align*}

So the two houses are 6 miles apart. Now convert miles to feet.

1 mile5280 feet=6 milesx feet\begin{align*}\frac{1 \ mile}{5280 \ feet} = \frac{6 \ miles}{x \ feet}\end{align*}

Now cross multiply to solve for x\begin{align*}x\end{align*}.

(1)xx=6(5280)=31,680\begin{align*}(1)x &= 6(5280)\\ x &= 31,680\end{align*}

The two houses are 31.680 feet apart.

In this example it would make sense to round up to 32 feet apart.

We can also work on problems involving scale models and measurement conversion. Let’s look at an example.

Example

A scale model of a building has a height of 3.5 feet. The scale of the model is 112 inch=10 feet\begin{align*}1 \frac{1}{2} \ inch = 10 \ feet\end{align*}. What is the actual height of the building?

The scale is in inches, but the scale model height is given in feet. First convert the scale height to inches. Then find the height of the building.

1 foot12 inches=3.5 feetx inches\begin{align*}\frac{1 \ foot}{12 \ inches} = \frac{3.5 \ feet}{x \ inches}\end{align*}

Now cross multiply to solve for x\begin{align*}x\end{align*}.

(1)xx=3.5(12)=42\begin{align*}(1)x &= 3.5(12)\\ x &= 42\end{align*}

So the height of the scale model is 42 inches. Now find the height of the actual building.

1.5 inch10 feet=42 inchesx feet\begin{align*}\frac{1.5 \ inch}{10 \ feet} = \frac{42 \ inches}{x \ feet}\end{align*}

Now cross multiply and solve for x\begin{align*}x\end{align*}.

(1.5)x1.5xx=42(10)=420=280\begin{align*}(1.5)x &= 42(10)\\ 1.5x &= 420\\ x &= 280\end{align*}

The actual building is 280 feet tall.

III. Solve Real – World Problems Involving Scaling of Customary Units of Measure

Now we can apply real – world problems to our work with customary units of measure. In this section, you will be using a scale to determine actual dimensions and scale dimensions. Be sure to convert customary units of measurement when necessary.

Example

Jeff ran his weekly long run of 13 miles in 2 hours. If his rate is 9.23 per mile, how long would it take Jeff to run 5280 feet?

To figure this out, you can use a proportion, but it might make more sense to think in terms of customary units of measurement. Here Jeff runs 9.23 per mile. How many feet are in one mile?

Yes, there are 5280 feet in one mile.

Therefore, Jeff runs 1 mile in 9.23.

Example

A scale model of a building has a scale of 2 inches for every 20 feet. If the size of the model is 2.5 feet, what is the actual height of the building?

First, write a proportion.

220 ft=2.5 ftx\begin{align*}\frac{2^{\prime \prime}}{20 \ ft} = \frac{2.5 \ ft}{x}\end{align*}

Notice that our inches and feet are all over the place. We have to rewrite the feet of the model measurement into inches. To do this, we multiply by 12.

2.5×12=30 inches\begin{align*}2.5 \times 12 = 30 \ inches\end{align*}

Now we can rewrite the proportion using the new measurements.

220 ft=30x\begin{align*}\frac{2^{\prime \prime}}{20 \ ft} = \frac{30^{\prime \prime}}{x}\end{align*}

Next, cross multiply and solve.

20×30=600\begin{align*}20 \times 30 = 600\end{align*}

The height of the building is 300 feet.

Now let’s go back to the problem from the introduction and work on solving it.

## Real-Life Example Completed

Everest in Miles

Here is the original problem once again. Reread it and then write a proportion with a variable to show how you would complete this conversion. Then solve the proportion for the number of miles. There are two parts to your answer.

“You keep telling me that Mount Everest is 29,035 feet high, but how many miles is that?” Teiyisha asked Josh during geography class.

“Well, you don’t need to know the miles because you are climbing it. Doesn’t it make more sense to measure it in feet?” Josh asked while working on his map.

“It might, but if I wanted to know how many miles, how could I figure this out?” Teiyisha asked.

Josh thought about it for a minute.

“You would have to do a conversion. You would have to change feet to miles, and we can do that if we know how many feet are in 1 mile. It still makes more sense to use feet though. Think about it, you can’t exactly draw a straight line up Mount Everest. Miles would be very hard to measure.”

“I get that, but I still want to know how many miles high it is,” Teiyisha explained.

“Alright, first let’s think about miles and feet. There are 5,280 feet in 1 mile.”

Now write a proportion and solve it. Be sure that your proportion has a variable in it. There are two parts to your answer.

Solution to Real – Life Example

First, we need to write a proportion to convert feet to miles. We know that there are 5,280 feet in 1 mile. This is the first part of the proportion. The second part of the proportion contains the unknown miles in a variable and the number of feet in Everest.

52801=29035x\begin{align*}\frac{5280}{1}=\frac{29035}{x}\end{align*}

Next, we cross multiply and divide to solve for the variable.

5280xx=29035=5.5 miles\begin{align*}5280x &= 29035\\ x &= 5.5 \ miles\end{align*}

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Customary System
the system of measurement that includes inches, feet, miles, pounds, tons, cups, quarts, gallons, etc.
Metric System
The system of measurement including centimeters and kiloliters that is used outside of the United States and often in the area of science.

## Time to Practice

Directions: Solve each problem by converting among customary units of measurement.

1. 102 inches = ______ feet
2. 25 pounds = ______ ounces
3. 160 cups = ______ gallons
4. 150 pounds = ______ tons
5. 6 feet = ______ inches
6. 360 inches = ______ feet
7. 5.5 feet = ______ inches
8. 900 inches = ______ feet
9. 32 ounces = ______ pounds
10. 320 ounces = ______ pounds
11. 6 pounds = ______ ounces
12. 15 pounds = ______ ounces
13. 6 cups = ______ pints
14. 3 gallons = ______ quarts
15. 8 quarts = ______ pints
16. 24 pints = ______ quarts

Directions: Solve each problem.

1. A recipe calls for 2 pints of milk. If Jorge cuts the recipe in half, how many cups of milk will he need?
2. Audrey is making brownies for a bake sale. The recipe calls for 8 ounces of flour for every 24 brownies. If she makes 96 brownies, how many pounds of flour will she need?
3. Two buildings are 5 inches apart on a map. The scale of the map is 14 inch=1 mile\begin{align*}\frac{1}{4} \ inch = 1 \ mile\end{align*}. What is the actual distance between the two buildings in yards?
4. The length of a classroom on a floor plan is 2.5 inches. The scale of the map is 12 inch=5 feet\begin{align*}\frac{1}{2} \ inch = 5 \ feet\end{align*}. What is the actual length of the classroom in inches?
5. A scale model of a mountain is 2.75 feet tall. The scale of the model is 14 inch=50 feet\begin{align*}\frac{1}{4} \ inch = 50 \ feet\end{align*}. What is the actual height of the mountain in feet?
6. A scale drawing of a town includes a park that measures 0.5 inch by 1.5 inches. If the scale of the map is 12 inch=1 mile\begin{align*}\frac{1}{2} \ inch = 1 \ mile\end{align*}, what is the area of the park in square feet?

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