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Scale Factor to Find Actual Dimensions

Use a ratio to find actual dimensions.

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Use Scale Factor When Problem Solving

Let’s Think About It

Credit: Josh Hallott
Source: https://www.flickr.com/photos/hyku/266895754/in/photolist-pzUNW-etb4V-9VUJdU-7jaBVy-duqdJb-7bcSaK-7uMdfL-2CGS5B-fUxjF-4For8o-anXWG1-gCLjtn-qHU9az-b82CEM-3Akkk5-nzqSH-doQGB8-7uM6nG-93Xf3y-8AeWeY-nny3eB-n3aDH-5PD75q-mX6kJn-23wxDT-oapoc3-7uM3BY-iYERAj-9fHRYM-7uHmmH-iYDx2r-9gRWNy-7uM5pW-7uM7ob-7uMagE-7uMbo3-6NmADm-7Zivw-oatyjX-jPM2R3-7uHiyi-23wwAg-72bR4-nqzd7r-ah8C1R-7oxWY7-bR6A-6Z7aob-9oGfAh-7Z2hPd
License: CC BY-NC 3.0

Lifan’s driveway has a length of 24 feet. If the scale is \begin{align*}2 \ \text{inches} : 4 \ \text{feet}\end{align*}, what is the scale factor? In a diagram, how many inches would Lifan draw to represent his driveway?

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

Guidance

The scale can be used to help you with scale dimensions or actual dimensions. This scale is key in problem solving.

If you look at the scale \begin{align*}2:1\end{align*}, you can use this information to determine the scale factor. The scale factor is the relationship between the scale dimension and the measurement comparison between the scale measurement of the model and the actual length. In this case the scale factor is \begin{align*}\frac{1}{2}\end{align*}.

Let’s look at an example.

What is the scale factor if 3 inches is equal to 12 feet?

First, write the ratio.

\begin{align*}\frac{3}{12}\end{align*}

Next, simplify the fraction.

\begin{align*}\frac{3}{12} = \frac{1}{4}\end{align*}

The answer is \begin{align*}\frac{1}{4}\end{align*}.

The scale factor is \begin{align*}1:4\end{align*}.

Now let’s look at a problem where you are applying this information.

If the scale dimension is 4, then you can figure out the actual dimension. Look at this proportion:

\begin{align*}1:2 = 4: x \end{align*}

First, put the proportion in fraction form.

\begin{align*}\frac{1}{2} = \frac{4}{x}\end{align*}  

Next, cross multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{1}{2} &=& \frac{4}{x} \\ 1x &=& 2 \times 4 \\ x &=& 8 \end{array}\end{align*}

The answer is 8.

This is the actual dimension.

Let’s look at a real world problem.

The plans for a flower garden show that it is 6 inches wide on the plan. If the scale for the flower garden is \begin{align*}1:12\end{align*}, what is the actual width of the flower garden?

First, write the proportion.

\begin{align*}1:12 = 6:x\end{align*} 

Next, put the proportion in fraction form.

\begin{align*}\frac{1}{12} = \frac{6}{x}\end{align*}  

Then, cross multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{1}{12} &=& \frac{6}{x} \\ 1x &=& 12 \times 6 \\ x &=& 72 \end{array}\end{align*}

The answer is 72.

The actual width of the flower garden is 72 inches.

Guided Practice

Find the missing actual dimension if the scale factor is \begin{align*}2^{{\prime}{\prime}}: 3^{\prime}\end{align*} and the scale measurement is \begin{align*}6^{{\prime}{\prime}}\end{align*}.

First, write the proportion.

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

Next, put the proportion in fraction form.

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

Then, cross multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{2}{3} &=& \frac{6}{x} \\ 2x &=& 3 \times 6 \\ 2x &=& 18 \end{array}\end{align*}

Then, divide both sides by 2 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 2x &=& 18 \\ \frac{2x}{2} &=& \frac{18}{2} \\ x &=& 9 \end{array}\end{align*}

The answer is 9.

The actual dimension is 9 feet.

Examples

Example 1

Find the missing actual dimension if the scale factor is \begin{align*}{\frac{1}{4}}^{{\prime}{\prime}}: 4^{\prime}\end{align*} and the scale measurement is \begin{align*}8^{{\prime}{\prime}}\end{align*}.

First, write the proportion.

\begin{align*}\frac{1}{4}:4 = 8:x\end{align*}

Next, put the proportion in fraction form.

\begin{align*}\begin{array}{rcl} \frac{\frac{1}{4}}{4} &=& \frac{8}{x} \\ \frac{1}{16} &=& \frac{8}{x} \end{array}\end{align*}

Then, cross multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{1}{16} &=& \frac{8}{x} \\ 1x &=& 8 \times 16 \\ x &=& 128 \end{array}\end{align*}

The answer is 128.

The actual dimension is 128 feet.

Example 2

Find the missing actual dimension if the scale factor is \begin{align*}{\frac{1}{4}}^{{\prime}{\prime}}:4^{\prime}\end{align*} and the scale measurement is \begin{align*}12^{\prime}\end{align*}.

First, write the proportion.

\begin{align*}\frac{1}{4}:4 = x:12\end{align*}

Next, put the proportion in fraction form.

\begin{align*}\begin{array}{rcl} \frac{\frac{1}{4}}{4} &=& \frac{x}{12} \\ \frac{1}{16} &=& \frac{x}{12} \end{array}\end{align*}

Then, cross multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{1}{16} &=& \frac{x}{12} \\ 16x &=& 1 \times 12 \\ 16x &=& 12 \end{array}\end{align*}

Then, divide both sides by 16 to solve for \begin{align*}x\end{align*} .

\begin{align*}\begin{array}{rcl} 16x &=& 12 \\ \frac{16x}{16} &=& \frac{12}{16} \\ x &=& \frac{12}{16} \\ x &=& \frac{3}{4} \end{array}\end{align*}

The answer is \begin{align*}\frac{3}{4}\end{align*}.

The actual dimension is \begin{align*}\frac{3}{4}\end{align*} inches.

Example 3

Find the missing actual dimension if the scale factor is \begin{align*}{\frac{1}{4}}^{{\prime}{\prime}}:4^{\prime}\end{align*} and the scale measurement is \begin{align*}16^{{\prime}{\prime}}\end{align*}.

First, write the proportion.

\begin{align*}\frac{1}{4}:4 = 16:x\end{align*}

Next, put the proportion in fraction form.

\begin{align*}\begin{array}{rcl} \frac{\frac{1}{4}}{4} &=& \frac{16}{x} \\ \frac{1}{16} &=& \frac{16}{x} \end{array}\end{align*}

Then, cross multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{1}{16} &=& \frac{16}{x} \\ 1x &=& 16 \times 16 \\ x &=& 256 \end{array}\end{align*}

The answer is 256.

The actual dimension is 256 feet.

Follow Up

Credit: Randy McRoberts
Source: https://www.flickr.com/photos/rmcrob/16226712059/in/photolist-qHU9az-etb4V-b82CEM-9VUJdU-3Akkk5-nzqSH-doQGB8-7uM6nG-7jaBVy-duqdJb-93Xf3y-8AeWeY-nny3eB-n3aDH-7bcSaK-7uMdfL-5PD75q-mX6kJn-23wxDT-7uHiyi-23wwAg-72bR4-oapoc3-nqzd7r-ah8C1R-7oxWY7-7uM3BY-bR6A-6Z7aob-iYERAj-9fHRYM-9oGfAh-7uHmmH-7Z2hPd-bLXq6x-aBodj2-iYDx2r-9RqoMD-66YCd-9gRWNy-6d4T2j-4vKd4v-9U8c4F-7uM5pW-4r4N4L-4HyrrU-7uM7ob-4xfhV1-aupKme-7v275E
License: CC BY-NC 3.0

Remember Lifan and his long driveway? The scale is \begin{align*}2 \text{ inches} : 4 \text{ feet}\end{align*} and the driveway is 24 feet long.

First, write the proportion. Note that \begin{align*}2:4\end{align*} is the scale factor.

\begin{align*}2:4 = x: 24\end{align*}

Next, put the proportion in fraction form.

\begin{align*}\frac{2}{4} = \frac{x}{24}\end{align*}

Then, cross multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{2}{4} &=& \frac{x}{24} \\ 4x &=& 2 \times 24 \\ 4x &=& 48 \end{array}\end{align*} 

Then, divide both sides by 4 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 4x &=& 48 \\ \frac{4x}{4} &=& \frac{48}{4} \\ x &=& 12 \end{array}\end{align*}

The answer is 12.

The scale dimension of Lifan’s driveway is 12 inches.

Video Review

https://www.youtube.com/watch?v=GC4aTrXNFJQ&feature=youtu.be

Explore More

Figure out each scale factor.

1. \begin{align*}\frac{2 \text{ inches}}{8 \text{ feet}}\end{align*}

2. \begin{align*}\frac{13 \text{ inches}}{12 \text{ feet}}\end{align*}

3. \begin{align*}\frac{6 \text{ inches}}{24 \text{ feet}}\end{align*}

4. \begin{align*}\frac{11 \text{ inches}}{33 \text{ feet}}\end{align*}

5. \begin{align*}\frac{16 \text{ inches}}{32 \text{ feet}}\end{align*}

6. \begin{align*}\frac{18 \text{ inches}}{36 \text{ feet}}\end{align*}

7. \begin{align*}\frac{6 \text{ inches}}{48 \text{ feet}}\end{align*}

8. \begin{align*}\frac{6 \text{ inches}}{12 \text{ feet}}\end{align*}

Solve each problem.

9. A rectangle has a width of 2 inches. A similar rectangle has a width of 9 inches. What scale factor could be used to convert the larger rectangle to the smaller rectangle?

10. A drawing of a man is 4 inches high. The actual man is 64 inches tall. What is the scale factor for the drawing?

11. A map has a scale of \begin{align*}1 \text{ inch}= 4 \text{ feet}\end{align*}. What is the scale factor of the map?

12. A drawing of a box has dimensions that are 2 inches, 3 inches, and 5 inches. The dimensions of the actual box will be \begin{align*}3\frac{1}{4}\end{align*} times the dimensions in the drawing. What are the dimensions of the actual box?

13. A room has a length of 10 feet. Hadley is drawing a scale drawing of the room, using the scale factor \begin{align*}\frac{1}{50}\end{align*}. How long will the room be in Hadley’s drawing?

14. The distance from Anna’s room to the kitchen is 15 meters. Anna is making a diagram of her house using the scale factor of \begin{align*}\frac{1}{150}\end{align*}. What will be the distance on the diagram from Anna’s room to the kitchen?

15. On a map of Cameron’s town, his house is 9 inches from his school. If the scale of the map is \begin{align*}\frac{1}{400}\end{align*}, what is the actual distance, in feet, from Cameron’s house to his school?

Vocabulary

Actual Dimension

Actual Dimension

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

Proportion

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

Scale Dimension

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

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