Dan is designing a map of a city. He has created a scale to represent the distance between objects on his map. The scale is 1" : 50'. Every inch of space on the map represents 50 feet of actual land.

There are two skyscrapers that are 9 inches apart on Dan's map. How far apart are the actual two buildings?

In this concept, you will learn about using a scale to determine actual measurements.

### Scale Measurements

Maps represent real places. Every part of the place has been reduced to fit on a single piece of paper. A map is an accurate representation because it uses a scale.

The **scale** is a ratio that relates the small size representation of a place to the actual size of a place.

Maps aren’t the only things that use scales. Architects use a scale when designing a house. A blueprint shows a small size of what the house will look like compared to the actual house. Any time an accurate model is built, it uses a scale.

Units of measurement are used to create a ratio that is the scale. The **ratio** compares two things. It compares the small size of the object or place to the actual size of the object or place.

A scale of 1 inch to 1 foot means that 1 inch on paper represents 1 foot in real space. This is how the ratio is written:

1” : 1'

If the distance between two points on a map is 2 inches, the scale tells us that the actual distance in is 2 feet.

Scales can be created in any size. One inch can represent 1,000 miles if the map shows a very large area, such as a continent. One centimeter might represent 1 meter if the map shows a small space, such as a room.

Scales are used in math to find the actual size of an object when only given the scale ratio.

For example, this scale below shows the ratio of

In other words, on the map, every \begin{align*}\frac{3}{4}''\end{align*} is equal to 20 miles.

The distance between two cities can be determined then by using this scale. Consider a map that shows two cities that are \begin{align*}\frac{3}{4}''\end{align*} apart. This would mean the actual distance between the cities is 20 miles per the scale.

Sometimes, the scale needs to be used to solve for a measurement that is not given directly in the scale. If the scale is 1”: 500 miles, how far is a city that measures \begin{align*}5\frac{1}{2}''\end{align*} on a map?

We know that every inch is 500 miles. We have \begin{align*}5\frac{1}{2}''\end{align*}. Let’s start with the 5.

\begin{align*}5 \times 500 = 2500 + \frac{1}{2} \times 500 = 2750\end{align*} mi.

Another way to do this is to write two ratios and cross multiply to solve for the missing information.

### Examples

#### Example 1

Earlier, you were given a problem about Dan and his map. He has a scale drawn on his map of 1" : 50 feet. Two buildings are represented on the map that are 9 inches apart. He needs to figure out how far apart those buildings are in actual scale.

First, Dan sets up the ratio proportion.

Next, Dan cross multiplies the equation.

\begin{align*}9in\cdot 50&= 1in\cdot x\\ 450&=1x\end{align*}

Then, solve for the missing information.

The answer is 450 feet.

Dan has figured out the two skyscrapers are 450 feet apart.

#### Example 2

If the scale is 2” : 1 ft, what is the actual measurement if a drawing shows the object as 6” long?

First, write a ratio that compares the scale.

\begin{align*}\frac{1 \ ft}{2''}=\frac{x \ ft}{6''}\end{align*}

The actual object size is unknown, so a variable is used to represent the unknown quantity.

Next, cross multiply the equation. In this problem, 1 ft. gets multiplied by 6 and x ft. gets multiplied by 2.

\begin{align*}1\cdot 6 &= 6\\ 2\cdot x &= 2x\\ 2x&=6 \end{align*}

Then, solve for the variable by isolating it.

\begin{align*}x&= 6\div 2\\ x&=3\end{align*}

The answer is that the object is actually 3 feet long.

#### Example 3

If the scale is 1” : 3 miles, how many miles does 5 inches represent?

First, write the ratio proportion.

Next, cross multiply the equation.

\begin{align*}3\cdot 5 = x \cdot 1\end{align*}

Then, solve for the missing information.

The answer is 15 miles.

#### Example 4

If the scale is 2” : 500 meters, how many meters does 4 inches represent?

First, write the ratio proportion.

\begin{align*}\frac{2in}{500m}=\frac{4in}{x}\end{align*}

Next, cross multiply the equation.

\begin{align*}4in\cdot 500m&= 2in\cdot x\\ 2000&= 2x\end{align*}

Then, solve for the missing information (x).

The answer is 1000 meters.

#### Example 5

If the scale is 5 in : 1000 feet, how many feet is 15 inches?

First, write the ratio proportion.

Next, cross multiply the equation.

\begin{align*}15 in\cdot 1000&= 5in\cdot x\\ 15000&=5x\end{align*}

Then, solve for the missing information.

The answer is 3,000 feet.

### Review

Use the given scale to determine the actual distance.

Given: 1” = 100 mi

- How many miles is 2” on the map?
- How many miles is \begin{align*}2\frac{1}{2} inch\end{align*} on the map?
- How many miles is \begin{align*}\frac{1}{4} inch\end{align*} on the map?
- How many miles is 8 inches on the map?
- How many miles is 16 inches on the map?
- How many miles is 12 inches on the map?
- How many miles is \begin{align*}\frac{1}{2} inch\end{align*} on the map?
- How many miles is \begin{align*}5 \frac{1}{4} inches\end{align*} on the map?

Given: 1 cm = 20 mi

- How many miles is 2 cm on the map?
- How many miles is 4 cm on the map?
- How many miles is 8 cm on the map?
- How many miles is 18 cm on the map?
- How many miles is 11 cm on the map?
- How many miles is \begin{align*}\frac{1}{2}\end{align*} cm on the map?
- How many miles is \begin{align*}1 \frac{1}{2}\end{align*} cm on the map?
- How many miles is \begin{align*}4 \frac{1}{4}\end{align*} cm on the map?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.7.