### Let’s Think About It

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?

In this concept, you will learn to convert customary units of measurement in real-world situations.

### Guidance

The **customary system**, also known as the Imperial System, is made up of units such as inches, feet, cups, gallons and pounds. Let’s look at conversions within the customary system of measurement.

**Customary Units of Measurement**

Let’s look at an example.

The distance from John’s house to Mike’s house on a map is 4.5 inches. The scale of the map is \begin{align*}1.5 \text{ inches}= 2 \text{ miles}\end{align*}. What is the actual distance from John’s house to Mike’s house in feet?

First, set up a proportion.

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

Next, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{1.5}{2} &=& \frac{4.5}{x} \\ 1.5x &=& 2 \times 4.5 \\ 1.5x &=& 9 \end{array} \end{align*}

Then, divide both sides by 1.5 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 1.5x &=& 9 \\ \frac{1.5x}{1.5}&=&\frac{9}{1.5} \\ x &=& 6 \end{array} \end{align*}

Then, if the actual distance is 6 miles, what is this distance in feet.

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

Then, cross multiply to solve for

.\begin{align*}\begin{array}{rcl} \frac{1}{5280} &=& \frac{6}{x} \\ 1x &=& 6 \times 5280 \\ x &=& 31680 \end{array}\end{align*}

The answer is 31680.

The distance between the two houses is 31,680 feet.

### Guided Practice

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

First, set up a proportion to find the height in inches.

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

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

\begin{align*}\begin{array}{rcl} \frac{12}{1} &=& \frac{x}{3.5} \\ 1x &=& 12 \times 3.5 \\ x &=& 42 \end{array}\end{align*}

The answer is 42.

The scale model is 42 inches high.

Then, set up a proportion to solve for the actual height of the building.

\begin{align*}\frac{1.5 \text{ inches}}{10 \text{ feet}} = \frac{42 \text{ inches}}{x \text{ feet}}\end{align*}

Then, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{1.5}{10} &=& \frac{42}{x} \\ 1.5x &=& 10 \times 42 \\ 1.5x &=& 420 \end{array}\end{align*}

Then, divide both sides by 1.5 in order to solve for

.\begin{align*}\begin{array}{rcl}
1.5x &=& 420 \\
\frac{1.5x}{1.5} &=& \frac{420}{1.5} \\
x &=& 280
\end{array}\end{align*}

The answer is 280.

The building is 280 feet tall.

### Examples

#### Example 1

Karin has a recipe that calls for 3 gallons of cider. How many quarts will she need?

First, set up a proportion.

\begin{align*}\frac{1\text{ gallon}}{4 \text{ quarts}} = \frac{3 \text{ gallons}}{x \text{ quarts}}\end{align*}

Next, cross multiply to solve for

.\begin{align*}\begin{array}{rcl}
\frac{1}{4} &=& \frac{3}{x} \\
1x &=& 3 \times 4 \\
x &=& 12
\end{array}
\end{align*}

The answer is 12.

Karin will need 12 quarts of cider.

#### Example 2

Jack threw the ball 12 feet. How many inches did he throw the ball?

First, set up a proportion.

\begin{align*}\frac{1 \text{ foot}}{12 \text{ inches}} = \frac{12 \text{ feet}}{x\text{ inches}}\end{align*}

Next, cross multiply to solve for

.\begin{align*}\begin{array}{rcl}
\frac{1}{12} &=&\frac{12}{x} \\
1x &=& 12 \times 12 \\
x &=& 144
\end{array}
\end{align*}

The answer is 144.

Jack threw the ball 144 inches.

#### Example 3

Carl drank 3 pints of lemonade. How many ounces did he drink?

First, set up a proportion.

\begin{align*}\frac{1 \text{ pint}}{16 \text{ ounces}} = \frac{3 \text{ pints}}{x \text{ ounces}}\end{align*}

Next, cross multiply to solve for

.\begin{align*}\begin{array}{rcl}
\frac{1}{16} &=& \frac{3}{x} \\
1x &=& 16 \times 3 \\
x &=& 48
\end{array}
\end{align*}

The answer is 48.

Carl drank 48 ounces of lemonade.

### Follow Up

Remember Evan and his thirsty conversion?

Evan needs to make 5 batches with each batch needing 3 cups of juice. He needs to find the total amount of juice in quarts.

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:

\begin{align*}3 \times 5 = 15 \text{ cups}\end{align*}

Next, set up a proportion.

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

Then, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{4}{1} &=& \frac{15}{x} \\ 4x &=& 1 \times 15 \\ 4x &=& 15 \end{array} \end{align*}

Then, divide both sides by 4 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl}
4x &=& 15 \\
\frac{4x}{4} &=& \frac{15}{4} \\
x &=& 3.75
\end{array}
\end{align*}

The answer is 3.75.

Evan needs to make 3.75 quarts of punch.

### Video Review

https://www.youtube.com/watch?v=4uQtJDf7drU&feature=youtu.be&safe=active

### Explore More

Solve each problem.

- Justin ran 3 miles. How many feet did he run?
- If the flour weighed four pounds, how many ounces did it weigh?
- How many pounds is equal to 4 tons?
- Mary needs 3 cups of juice for a recipe. How many ounces does she need?
- Jess bought 3 quarts of pineapple juice. How many pints did she purchase?
- If Karen bought 16 quarts of ice cream, how many gallons did she buy?
- The length of the garden is four yards. How many feet is that?
- If the width of the garden is 4 yards, how many inches is that?
- Will eight cups of water fit in a two quart saucepan?
- A recipe calls for 2 pints of milk. If Jorge cuts the recipe in half, how many cups of milk will he need?
- 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?
- Two buildings are 5 inches apart on a map. The scale on the map is \begin{align*}\frac{1}{4} \text{ inch}=1 \text{ mile}\end{align*}. What is the actual distance between the two buildings?
- The length of a classroom on a floor plan is 2.5 inches. The scale of the map is \begin{align*}\frac{1}{2} \text{ inches}= 5 \text{ feet}\end{align*} What is the actual length of the classroom in inches?
- A scale model of a mountain is 2.75 feet tall. The scale of the model is What is the actual height of the mountain in feet?
- 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 \begin{align*}0.5 \text{ inches} = 1\text{ mile}\end{align*}, what is the area of the park in square feet?