### Let's Think About It

Tanya's room was a mess. She had all sorts of books and toys strewn about, and her grandpa decided it was time to do something about it. He asked Tanya to draw a diagram for him, and he would make bookcase shelves. Tanya's drawing shows that the bookcase is 5.5 inches high. The scale says, 2 inches = 1 foot.

Given this scale, what is the actual height of the bookcase?

In this concept, you will learn how to use a unit scale to find actual dimensions.

### Guidance

A **ratio** compares two quantities. Ratios can be written as fractions, with a colon or with the word “to.”

\begin{align*}\frac{2}{3}\end{align*}, 2:3, and "2 to 3" are ratios.

A **proportion** is created when two ratios are found to be equivalent or equal.

\begin{align*}\frac{3}{6}\end{align*} is a proportion.

=A **unit rate** is a comparison of two measurements, one of which has a value of 1.

55 miles per hour, \begin{align*}\frac{55}{1}\frac{miles}{hour}\end{align*} is a unit rate.

A **unit scale** is a ratio that compares the dimensions of an actual object to the dimensions of a scale drawing or model that represents the actual object. Neither value in a unit scale has to equal 1.

The unit scale on a map may read 1" = 100 feet.

** **The ratio would be written .

To represent a line 500 feet long, the unit scale would be used to draw a line 5 inches long.

A line 8 inches long would represent an actual line of 800 feet.

Unit scales and proportions can be used to calculate actual distances from maps, drawings, or models.

Actual distances can be represented on maps, drawings, or models by using unit scales.

### Guided Practice

Hector made this scale model of the Statue of Liberty. The scale height of his model, from the base to the torch, is 4.65 centimeters. Find the actual height of the Statue of Liberty if the scale is 1 cm = 10 m.

First, write the unit scale as a ratio.

\begin{align*}\frac{centimeters}{meters} = \frac{1}{10}\end{align*}

Next, keeping the units consistent, write a ratio that compares the scale height to the actual height.

\begin{align*}\frac{4.65}{h}=\frac{centimeters}{meters}\end{align*}

Then, set the ratios equal to one another to form a proportion.

Next, cross multiply.

\begin{align*}h = 46.5 \ meters\end{align*}

The answer is that the actual height of the Statue of Liberty from the base to the torch is 46.5 meters.

### Examples

#### Example 1

A beetle that Amelia observed was too small to draw at its actual size. So, she made this scale drawing.

In her drawing, the length of the beetle was 4.8 cm. What was the length of the actual beetle?

First, write the unit scale as a ratio.

\begin{align*}\frac{centimeters}{millimeters} = \frac{0.4}{1}\end{align*}

Next, keeping the units consistent, write a ratio that compares the scale size to the actual size.

\begin{align*}\frac{4.8}{x} =\frac{centimeters}{millimeters}\end{align*}

Then, set the ratios equal to one another to form a proportion.

Next, cross multiply.

The answer is that the beetle is 12 millimeters long.

#### Example 2

A map has a scale of 1" = 10 miles. How far away is a city that measures 12 inches on the map?

First, write the unit scale as a ratio.

\begin{align*}\frac{inch}{miles} = \frac{1}{10}\end{align*}

Next, keeping the units consistent, write a ratio that compares the scale distance to the actual distance.

\begin{align*}\frac{12}{d}=\frac{in}{mi}\end{align*}

Then, set the ratios equal to one another to form a proportion.

Next, cross multiply.

The answer is that the city is 120 miles away.

#### Example 3

The map below shows the distances between three towns. On the map, the distance between Smithville and Frankton is

inches. Find the actual straight-line distance between Smithville and Frankton.

First, write the unit scale as a ratio.

\begin{align*}\frac{inches}{miles} = \frac{0.25}{2}\end{align*}

Next, keeping the units consistent, write a ratio that compares the scale distance to the actual distance.

\begin{align*}\frac{2.25}{d}\end{align*}

Then, set the ratios equal to one another to form a proportion.

Next, cross multiply.

The answer is that the actual distance from Smithville and Frankton is 18 miles.

### Follow Up

Remember Tanya and her messy room?

She made a scale drawing so her grandpa could build shelves. Tanya's drawing shows that the bookcase is \begin{align*}5.5\end{align*} inches high. The scale says, 2 inches = 1 foot. Give the actual height of the shelf.

First, write the unit scale as a ratio.

\begin{align*}\frac{inches}{feet} = \frac{2}{1}\end{align*}

Next, keeping the units consistent, write a ratio that compares the scale distance to the actual distance.

\begin{align*}\frac{5.5}{h}\end{align*}

Then, set the ratios equal to one another to form a proportion.

Next, cross multiply

The answer is that the actual height of the book shelf is 2.75 feet.

### Video Review

### Explore More

Use the given scale to determine the actual distance.

Given: Scale 1” = 100 miles

1. How many miles is 2” on the map?

2. How many miles is \begin{align*}2\frac{1}{2} inch\end{align*} on the map?

3. How many miles is \begin{align*}\frac{1}{4} inch\end{align*} on the map?

4. How many miles is 10 inches on the map?

5. How many miles is 11 inches on the map?

6. How many miles is 12.5 inches on the map?

7. How many miles is \begin{align*}\frac{1}{2}\end{align*} inch on the map?

8. How many miles is \begin{align*}5 \frac{1}{4}\end{align*} inch on the map?

Given: 1 cm = 20 mi

9. How many miles is 2 cm on the map?

10. How many miles is 4 cm on the map?

11. How many miles is 8.5 cm on the map?

12. How many miles is 19 cm on the map?

13. How many miles is 13 cm on the map?

14. How many miles is \begin{align*}\frac{1}{2}\end{align*} cm on the map?

15. How many miles is \begin{align*}1 \frac{1}{2}\end{align*} cm on the map?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 5.12.