# Distances or Dimensions Given Scale Measurements

## Identify actual dimensions given scale dimensions

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Distances or Dimensions Given Scale Measurements

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 34":20\begin{align*}\frac{3}{4}" : 20\end{align*}mi.

In other words, on the map, every  34′′\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  34′′\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 512′′\begin{align*}5\frac{1}{2}''\end{align*} on a map?

We know that every inch is 500 miles. We have 512′′\begin{align*}5\frac{1}{2}''\end{align*}. Let’s start with the 5.

5×500=2500+12×500=2750\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.

1in50ft=9inx\begin{align*}\frac{1in}{50ft}=\frac{9in}{x}\end{align*}

Next, Dan cross multiplies the equation.

9in50450=1inx=1x\begin{align*}9in\cdot 50&= 1in\cdot x\\ 450&=1x\end{align*}

Then, solve for the missing information.

450=x\begin{align*}450=x\end{align*}

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.

1 ft2′′=x ft6′′\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.

162x2x=6=2x=6\begin{align*}1\cdot 6 &= 6\\ 2\cdot x &= 2x\\ 2x&=6 \end{align*}

Then, solve for the variable by isolating it.

xx=6÷2=3\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.

1"3mi=5inxmi.\begin{align*}\frac{1"}{3 mi}=\frac{5 in}{x mi.}\end{align*}

Next, cross multiply the equation.

35=x1\begin{align*}3\cdot 5 = x \cdot 1\end{align*}

Then, solve for the missing information.

15=x\begin{align*}15= x\end{align*}

#### Example 4

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

First, write the ratio proportion.

2in500m=4inx\begin{align*}\frac{2in}{500m}=\frac{4in}{x}\end{align*}

Next, cross multiply the equation.

4in500m2000=2inx=2x\begin{align*}4in\cdot 500m&= 2in\cdot x\\ 2000&= 2x\end{align*}

Then, solve for the missing information (x).

2000÷21000=x=x\begin{align*}2000\div 2&=x\\ 1000 &=x\end{align*}

#### Example 5

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

First, write the ratio proportion.

5in1000ft=15inx\begin{align*}\frac{5in}{1000ft}=\frac{15in}{x}\end{align*}

Next, cross multiply the equation.

15in100015000=5inx=5x\begin{align*}15 in\cdot 1000&= 5in\cdot x\\ 15000&=5x\end{align*}

Then, solve for the missing information.

15000÷53000=x=x\begin{align*}15000\div 5&=x\\ 3000&=x\end{align*}

### Review

Use the given scale to determine the actual distance.

Given: 1” = 100 mi

1. How many miles is 2” on the map?
2. How many miles is 212inch\begin{align*}2\frac{1}{2} inch\end{align*} on the map?
3. How many miles is 14inch\begin{align*}\frac{1}{4} inch\end{align*} on the map?
4. How many miles is 8 inches on the map?
5. How many miles is 16 inches on the map?
6. How many miles is 12 inches on the map?
7. How many miles is 12inch\begin{align*}\frac{1}{2} inch\end{align*} on the map?
8. How many miles is 514inches\begin{align*}5 \frac{1}{4} inches\end{align*} on the map?

Given: 1 cm = 20 mi

1. How many miles is 2 cm on the map?
2. How many miles is 4 cm on the map?
3. How many miles is 8 cm on the map?
4. How many miles is 18 cm on the map?
5. How many miles is 11 cm on the map?
6. How many miles is 12\begin{align*}\frac{1}{2}\end{align*} cm on the map?
7. How many miles is 112\begin{align*}1 \frac{1}{2}\end{align*} cm on the map?
8. How many miles is 414\begin{align*}4 \frac{1}{4}\end{align*} cm on the map?

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### Vocabulary Language: English

TermDefinition
Inches An inch is a customary unit of measurement, measured best by a ruler.
Length Length is a measurement of how long something is. Examples of customary units of length are inches, feet, yards and miles.
Proportion A proportion is an equation that shows two equivalent ratios.
Scale Scale is the relationship between the size of a drawing and the size of the real object.
Two – Dimensional A figure drawn in two dimensions is only drawn using length and width.