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# 4.6: Use Unit Scale When Problem Solving

Difficulty Level: At Grade Created by: CK-12
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Practice Unit Scale to Find Actual Dimensions

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Josh is building a scale diagram of Mount Everest. He knows that Mount Everest is 29,035 feet high. He doesn’t want his model to be too large so thinks he should use \begin{align*}\frac{1 \ in}{2000 \ ft}\end{align*}. If Josh uses this scale, will his model be too big?

In this concept, you will learn to use unit scale when problem solving.

### Unit Scale

Sometimes in life, you have a real-life object that you want to represent in a smaller form. Think about buildings. When an architect is planning a building design, he or she may design a model of the building. This model shows the dimensions of the building in a smaller way. When you do this, you take the actual dimensions and shrink them down to build a model. When you do this, you create a unit scale for the model. When you create a unit scale, you decide on an appropriate measurement to represent an actual measurement.

First, let’s look at a unit scale.

\begin{align*}1 \ \text{inch} = 3 \ \text{feet}\end{align*}

This is a unit scale. You have a unit represented by the one inch. Remember that when you talk about unit, you are talking about a relationship to one. You have one inch represented by three feet.

The one inch is the scale dimension and the three feet is the actual dimension we are measuring.

Now, not all objects that you will create a model of will measure exactly what the unit scale does, so you have to use a unit scale to show the relationship between scale dimensions and actual dimensions. Scale dimensions are the dimensions of the model, and actual dimensions are the real–life dimensions.

Let’s look at an example.

Using the unit scale above, what would be the relationship between the scale dimensions and the actual dimensions for an object 24 feet long?

\begin{align*}1 \ \text{inch} = 3 \ \text{feet}\end{align*}

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.

\begin{align*} \frac{1 \ \text{in}}{3 \text{ ft}}=\frac{x \ \text{in}}{24 \ \text{ft}}\end{align*}Then, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{1 \ \text{in}}{3 \text{ ft}} &=& \frac{x \ \text{in}}{24 \ \text{ft}} \\ 3x &=& 1 \times 24 \\ 3x &=& 24 \end{array}\end{align*}

Then, divide both sides by 3 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 3x &=& 24 \\ \frac{3x}{3} &=& \frac{24}{3} \\ x &=& 8 \end{array}\end{align*}

Therefore, \begin{align*}8 \ \text{inches}= 24 \ \text{feet}\end{align*}.

If you know the scale dimensions and the unit scale, then you can find the actual dimensions with the unit scale.

If you know the actual dimensions and the unit scale, then you can find the scale dimensions with the unit scale.

Let’s look at an example.

What is the scale length of the object if the unit scale is 2 inches : 4 feet and the actual dimensions of the object is 20 feet?

\begin{align*}2 \ \text{inches} = 4 \ \text{feet}\end{align*}

Next, use the scale dimensions to write a proportion of the scale dimensions to the actual dimensions.

\begin{align*}\frac{2 \ \text{in}} {4 \ \text{ft}}=\frac{x \ \text{in}} {20 \ \text{ft}}\end{align*}

Then, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{2 \ \text{in}} {4 \ \text{ft}}&=& \frac{x \ \text{in}} {20 \ \text{ft}} \\ 4x &=& 2 \times 20 \\ 4x &=& 40 \end{array}\end{align*}

Then, divide both sides by 4 to solve for \begin{align*}x\end{align*}.  \begin{align*}\begin{array}{rcl} 4x &=& 40 \\ \frac{4x}{4} &=& \frac{40}{4} \\ x &=& 10 \end{array}\end{align*}The answer is 10.

Therefore, \begin{align*}10 \ \text{inches} = 20 \ \text{feet}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Josh’s mountainous model.

Josh is using scale dimensions of \begin{align*}1\ \text{in} = 2000\ \text{ft}\end{align*}. Mount Everest is 29,035 feet.

\begin{align*}1\ \text{inch} = 2000 \ \text{feet}\end{align*}

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.

\begin{align*}\frac{1 \ \text{in}}{2000 \ \text{ft}}=\frac{x \ \text{in}}{29035 \ \text{ft}}\end{align*}

Then, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{1 \ \text{in}}{2000 \ \text{ft}} &=& \frac{x \ \text{in}}{29,035 \ \text{ft}} \\ 2000x &=& 1 \times 29,035 \\ 2000x &=& 29,035 \end{array}\end{align*}

Then, divide both sides by 2000 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 2000x &=& 29,035 \\ \frac{2000x}{2000} &=& \frac{29,035}{2000} \\ x &=& 14.5 \end{array}\end{align*}

Josh’s model of Mount Everest would be 14.5 inches tall.

#### Example 2

Using a unit scale of 1 inch : 8 feet, what is the actual dimension of an object with a scale dimension for length of 5 inches?

\begin{align*}1 \ \text{inch} = 8 \ \text{feet}\end{align*}

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.

\begin{align*}\frac{1 \ \text{in}}{8 \ \text{ ft}}=\frac{5 \ \text{in}}{x \ \text{ft}} \end{align*}Then, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{1 \ \text{in}}{8 \ \text{ ft}} &=& \frac{5 \ \text{in}}{x \ \text{ft}} \\ 1x &=& 5 \times 8 \\ x &=& 40 \end{array}\end{align*}The answer is 40.

Therefore, the actual dimensions are \begin{align*}5 \ \text{inches }= 40 \ \text{feet}\end{align*}.

#### Example 3

Using a unit scale of 1 inch : 5 feet, what is the actual dimension of an object with a scale dimension for length of 25 feet?

\begin{align*}1 \ \text{inch}= 5 \ \text{feet}\end{align*}

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.

\begin{align*}\frac{1 \ \text{in}}{5 \ \text{ft}}=\frac{x \ \text{in}}{25 \ \text{ft}} \end{align*}Then, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{1 \ \text{in}}{5 \ \text{ft}} &=& \frac{x \ \text{in}}{25 \ \text{ft}} \\ 5x &=& 1 \times 25 \\ 5x &=& 25 \end{array}\end{align*}Then, divide both sides by 5 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 5x &=& 25 \\ \frac{5x }{5} &=& \frac{25}{5} \\ x &=& 5 \end{array}\end{align*}

The actual dimensions are \begin{align*}5 \ \text{inches} = 25 \ \text{feet}\end{align*}.

#### Example 4

Using a unit scale of 1 inch : 5 feet, what is the actual dimension of an object with a scale dimension for length of 3 inches?

\begin{align*}1 \ \text{inch} = 5 \ \text{feet}\end{align*}

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.

\begin{align*}\frac{1 \ \text{in}}{5 \ \text{ft}}=\frac{3 \ \text{in}}{x \ \text{ft}}\end{align*}

Then, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{1 \ \text{in}}{5 \ \text{ft}} &=& \frac{3 \ \text{in}}{x \ \text{ft}} \\ 1x &=& 3 \times 5 \\ x &=& 15 \end{array}\end{align*}

The actual dimensions are \begin{align*}3 \ \text{inches}= 15 \ \text{feet}\end{align*}.

#### Example 5

Using a unit scale of 1 inch : 5 feet, what is the actual dimension of an object with a scale dimension for length of 75 feet?

\begin{align*}1 \ \text{inch} = 5 \ \text{feet}\end{align*}

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.

\begin{align*}\frac{1 \ \text{in}}{5 \ \text{ft}}=\frac{x \ \text{in}}{75 \ \text{ft}}\end{align*}

Then, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{1 \ \text{in}}{5 \ \text{ft}} &=& \frac{x \ \text{in}}{75 \ \text{ft}} \\ 5x &=& 1 \times 75 \\ 5x &=& 75 \end{array}\end{align*}

Then, divide both sides by 5 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 5x &=& 75 \\ \frac{5x} {5} &=& \frac{75}{5} \\ x &=& 15 \end{array}\end{align*}

The actual dimensions are \begin{align*}15 \ \text{inches} = 75 \ \text{feet}\end{align*}.

### Review

Find the scale dimension given the scale. Write a proportion and an answer for each problem. There are two answers for each problem.

1. Scale is \begin{align*}1^{\prime \prime }= 2 \ \text{ft}\end{align*}, the actual dimension is 18 feet

2. Scale is \begin{align*}1^{\prime \prime}= 5 \ \text{feet}\end{align*}, the actual dimension is 20 feet

3. Scale is \begin{align*}1^{\prime \prime}= 2 \ \text{feet}\end{align*}, actual dimension is 10 feet

4. Scale is \begin{align*}1^{\prime \prime}= 12 \ \text{feet}\end{align*}, actual dimension is 72 feet

5. Scale is \begin{align*}3^{\prime \prime}= 4 \ \text{feet}\end{align*}, actual dimension is 16 feet

Using a scale of 1 to 2, figure out the actual dimensions given each scale.

6. 4 to \begin{align*}\underline{\;\;\;\;\;\;\;\;}\end{align*}

7. 6 to \begin{align*}\underline{\;\;\;\;\;\;\;\;}\end{align*}

8. 9 to \begin{align*}\underline{\;\;\;\;\;\;\;\;}\end{align*}

9. 12 to \begin{align*}\underline{\;\;\;\;\;\;\;\;}\end{align*}

10. 14 to \begin{align*}\underline{\;\;\;\;\;\;\;\;}\end{align*}

Using a scale of 3 to 4, figure out the actual dimensions given each scale.

11. 6 to \begin{align*}\underline{\;\;\;\;\;\;\;\;}\end{align*}

12. 9 to \begin{align*}\underline{\;\;\;\;\;\;\;\;}\end{align*}

13. 12 to \begin{align*}\underline{\;\;\;\;\;\;\;\;}\end{align*}

14. 18 to \begin{align*}\underline{\;\;\;\;\;\;\;\;}\end{align*}

15. 36 to \begin{align*}\underline{\;\;\;\;\;\;\;\;}\end{align*}

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Color Highlighted Text Notes

### Vocabulary Language: English

Actual Dimension

The actual dimensions are the real–life measures of the object or building.

Proportion

A proportion is an equation that shows two equivalent ratios.

Scale Dimension

A scale dimension is the measurement used to represent actual dimensions in a drawing or on a map.

Unit Scale

The unit scale is the scale of measurement used to represent actual dimensions in a model or drawing. The scale includes units of measurement such as inches, feet, meters.

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