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2.8: Scale Distances or Dimensions

Difficulty Level: At Grade Created by: CK-12
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A city planner is drawing a map of the community park. The park spans 320 feet along the main street of the city. The city planner needs to represent this park accurately drawn to a scale of 2 cm:10 feet. How long in centimeters should the drawing of the park be on the map?

In this concept, you will learn how to determine scale dimensions when given actual dimensions.

Scale Dimensions

Scales are used in math to represent larger objects on maps or diagrams. You may have learned how to take a scale and determine the actual size of a large object. On the other hand, you can determine the size of the smaller representation when given and scale and the size of the actual objects being measures.

To make a map, for instance, you “shrink” actual distances down to a smaller size that can be shown on a piece of paper. A scale is used, but instead of solving for the actual distance, you solve for the map distance (the smaller representation).

Suppose you are making a map of some nearby towns. City A and City B are 350 kilometers apart in real distance. The scale for the map is 1 cm : 10 km. How far apart would City A and City B be on the map drawn to scale?

The scale is used to write ratios that make a proportion. Then, the given information is filled in. The actual distance between the two towns is given, so that information is plugged into the proportion and the map distance is what is being solved for.

1 cm10 km=x cm350 km\begin{align*}\frac{1 \ cm}{10 \ km}=\frac{x \ cm}{350 \ km}\end{align*}

Cross multiply to find the number of centimeters that would need to be drawn on the map.

1(350)35035=10x=10x=x\begin{align*}1(350) &= 10x\\ 350 &= 10x\\ 35 &= x\end{align*}

The scale was used to determine that City A and City B should be drawn 35 cm apart on the map to represent the actual 350 km apart in real life.

This concept can be applied to finding the scale of smaller representations of objects too.

Jesse wants to build a model of a building. The building is 100 feet tall. If Jesse wants to use a scale of 1” : 25 feet, how tall will his model be?

Start by looking at the scale and write a proportion to show the measurements that we know.

1′′25 ft=x100 ft\begin{align*}\frac{1''}{25 \ ft}=\frac{x}{100 \ ft}\end{align*}

To solve this proportion, cross multiply.

1(100)1004=25(x)=25x=x\begin{align*}1(100) &= 25(x)\\ 100 &= 25x\\ 4 &= x\end{align*}

Jesse’s model will be 4 inches tall.

The answer is 4′′\begin{align*}4''\end{align*}.

Examples

Example 1

Earlier, you were given a problem about the city planner drawing the map of the community park.

The scale is 2 cm:10 feet and the actual length of the park is 320 feet along the main street. The city planner needs to use ratios and proportions to solve for the scaled down dimension of the park to be drawn on the map.

First, the city planner sets up a proportion using the given information.

2cm10ft=x320ft\begin{align*}\frac{2cm}{10ft}=\frac{x}{320ft}\end{align*}

Next, he cross multiplies the equation to be able to solve for the missing information.

3202640=10x=10x\begin{align*}320 \cdot 2&= 10\cdot x\\ 640&=10x\end{align*}

Then, the city planner isolates the variable to solve for the scale size of the park on the map.

640÷1064=x=x\begin{align*}640\div 10&=x\\ 64&=x\end{align*}

The answer is 64 cm. The city planner will draw the park to be 64 cm long to accurately represent the scale on the map.

Example 2

Find the scale size in inches of an object that is 480 ft tall when the scale is 1" : 10 ft.

First, write the proportion.

1in10ft=x480ft\begin{align*}\frac{1in}{10ft}=\frac{x}{480ft}\end{align*}

Next, cross multiply to solve for the missing information.

480ft1480=10ftx=10x\begin{align*}480ft \cdot 1&=10ft\cdot x\\ 480&=10x\end{align*}

Then, solve for x by isolating the variable

480÷1048=x=x\begin{align*}480\div 10&=x\\ 48&=x\end{align*}

The answer is 48 inches. The object that is actually 480 feet tall would be represented in a 48 inch model to scale.

Example 3

The distance from Kara's home to the family summer house is 150 miles. How many inches is that on the map?

First, write the proportion.

1in100mi=x150mi\begin{align*}\frac{1in}{100mi}=\frac{x}{150mi}\end{align*}

Next, cross multiply.

1501150=100x=100x\begin{align*}150 \cdot 1&= 100\cdot x\\ 150&=100x\end{align*}

Then, isolate the variable "x" to solve the problem.

150÷1001.5=x=x\begin{align*}150\div 100&=x\\ 1.5&=x\end{align*}

Example 4

The distance from Kara's home to her Grandmother's home is 2000 miles. How many inches is that on the map?

First, write the proportion.

1in100mi=x2000mi\begin{align*}\frac{1in}{100mi}=\frac{x}{2000mi}\end{align*}

Next, cross multiply.

200012000=100x=100x\begin{align*}2000\cdot 1&=100\cdot x\\ 2000&=100x\end{align*}

Then, isolate the variable to solve.

2000÷10020=x=x\begin{align*}2000\div 100&=x\\ 20&=x\end{align*}

Example 5

If the distance from Mark's home to Kara's is 40 miles, how many inches is that on the map?

First, write the proportion.

1in100mi=x40mi\begin{align*}\frac{1in}{100mi}=\frac{x}{40mi}\end{align*}

Next, cross multiply.

40140=100x=100x\begin{align*}40\cdot 1&= 100\cdot x\\ 40&=100x\end{align*}

Then, isolate the variable to solve.

40÷100.4=x=x\begin{align*}40\div 100&=x\\ .4&=x\end{align*}

Review

Use the given scale to determine the scale measurement given the actual distance.

Given Scale 2” = 150 miles

1. How many scale inches would 300 miles be?
2. How many scale inches would 450 miles be?
3. How many scale inches would 75 miles be?
4. How many scale inches would 600 miles be?
5. How many scale inches would 900 miles be?

Use the given scale to determine the scale measurement for the following dimensions.

Given Scale 1” = 1 foot

1. What is the scale measurement for a room that is 8’ ×\begin{align*}\times\end{align*} 12’?
2. What is the scale measurement for a tree that is 1 yard high?
3. What is the scale measurement for a tower that is 36 feet high?
4. How many feet is that?
5. What is the scale measurement for a room that is 12×1612\begin{align*}12' \times 16 \frac{1}{2}'\end{align*}?

Use what you have learned about scale and measurement to answer each of the following questions.

1. Joaquin is building the model of a tower. He is going to use a scale of 1” = 1 foot. How big will his tower be in inches if the actual tower if 480 feet tall?
2. How many feet high will the model be?
3. Is this a realistic scale for this model? Why or why not?
4. If Joaquin decided to use a scale that was 14′′\begin{align*}\frac{1}{4}''\end{align*} for every 1 foot, how many feet high would his model be?
5. What scale would Joaquin need to use if he wanted his model to be 5 feet tall?
6. How tall would the model be if Joaquin decided to use 116′′=1\begin{align*}\frac{1}{16}'' = 1\end{align*} foot?

Vocabulary Language: English

Proportion

Proportion

A proportion is an equation that shows two equivalent ratios.
Scale

Scale

Scale is the relationship between the size of a drawing and the size of the real object.

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Date Created:
Oct 29, 2012