# 4.8.4: Find the Scale Factor

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## Use Scale Factor When Problem Solving

Lifan’s driveway has a length of 24 feet. If the scale is 2 inches:4 feet, 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.

**Scale Factor**

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 2:1, 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 1/2.

Let’s look at an example.

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

First, write the ratio.

3/12

Next, simplify the fraction.

3/12=1/4

The answer is 1/4.

The scale factor is 1:4.

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**:

1:2=4:x

First, put the proportion in fraction form.

1/2=4/x

Next, cross multiply to solve for x.

1/2=4/x

1x=2×4

x=8

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 1:12, what is the actual width of the flower garden?

First, write the proportion.

1:12=6:x

Next, put the proportion in fraction form.

1/12=6/x

Then, cross multiply to solve for x.

1/12=6/x

1x=12×6

x=72

The answer is 72.

The actual width of the flower garden is 72 inches.

**Examples**

Example 4.8.4.1

Earlier, you were given a problem about Lifan and his long driveway. The scale is 2 inches:4 feet and the driveway is 24 feet long.

**Solution**

First, write the proportion. Note that 2:4 is the scale factor.

2:4=x:24

Next, put the proportion in fraction form.

2/4=x/24

Then, cross multiply to solve for x.

2/4=x/24

4x=2×24

4x=48

Then, divide both sides by 4 to solve for x.

4x=48

4x/4=484

x=12

The answer is 12.

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

Example 4.8.4.2

Find the missing actual dimension if the scale factor is 2′′:3′ and the scale measurement is 6′′.

**Solution**

First, write the proportion.

2:3=6:x

Next, put the proportion in fraction form.

2/3=6/x

Then, cross multiply to solve for x.

2/3=6/x

2x=3×6

2x=18

Then, divide both sides by 2 to solve for x.

2x=18

2x/2=18/2

x=9

The answer is 9.

The actual dimension is 9 feet.

Example 4.8.4.3

Find the missing actual dimension if the scale factor is 1/4′′:4′ and the scale measurement is 8′′.

**Solution**

First, write the proportion.

1/4:4=8:x

Next, put the proportion in fraction form.

1/4/4=8/x

1/16=8/x

Then, cross multiply to solve for x.

1/16=8/x

1x=8×16

x=128

The answer is 128.

The actual dimension is 128 feet.

Example 4.8.4.4

Find the missing actual dimension if the scale factor is 1/4′′:4′ and the scale measurement is 12′.

**Solution**

First, write the proportion.

1/4:4=x:12

Next, put the proportion in fraction form.

1/4/4=x/12

1/16=x/12

Then, cross multiply to solve for x.

1/16=x/12

16x=1×12

16x=12

Then, divide both sides by 16 to solve for x .

16x=12

16x/16=12/16

x=12/16

x=3/4

The answer is 3/4.

The actual dimension is 3/4 inches.

Example 4.8.4.5

Find the missing actual dimension if the scale factor is 1/4′′:4′ and the scale measurement is 16′′.

**Solution**

First, write the proportion.

1/4:4=16:x

Next, put the proportion in fraction form.

1/4/4=16/x

1/16=16/x

Then, cross multiply to solve for x.

1/16=16/x

1x=16×16

x=256

The answer is 256.

The actual dimension is 256 feet.

**Review**

Figure out each scale factor.

1. 2 inches/8 feet

2. 13 inches/12 feet

3. 6 inches/24 feet

4. 11 inches/33 feet

5. 16 inches/32 feet

6. 18 inches/36 feet

7. 6 inches/48 feet

8. 6 inches/12 feet

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 1 inch=4 feet. 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 3(1/4) 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 1/50. 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 1/150. 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 1/400, what is the actual distance, in feet, from Cameron’s house to his school?

**Review (Answers)**

To see the Review answers, open this PDF file and look for section 4.7.

**Vocabulary**

Term | Definition |
---|---|

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. |