Scale factor worksheet: Fill out & sign online | DocHub - Free Printable

To solve the problems in the worksheet, we need to use the concept of scale factors. The scale factor is a ratio that relates the dimensions of a model or map to the actual dimensions. We will calculate the missing values step by step.

Problem 1:

The scale factor for a model is \( 5 \, \text{cm} = \_\_\_\_\_ \, \text{m} \).

- Model: \( 9.5 \, \text{cm} \)
- Actual: \( 30.5 \, \text{m} \)

#### Solution:
The scale factor is given by:
\[
\text{Scale factor} = \frac{\text{Actual dimension}}{\text{Model dimension}}
\]
\[
\text{Scale factor} = \frac{30.5 \, \text{m}}{9.5 \, \text{cm}}
\]

Convert \( 9.5 \, \text{cm} \) to meters:
\[
9.5 \, \text{cm} = 0.095 \, \text{m}
\]

Now calculate:
\[
\text{Scale factor} = \frac{30.5}{0.095} \approx 321.05
\]

Thus, \( 5 \, \text{cm} \) corresponds to:
\[
5 \, \text{cm} \times 321.05 = 1605.25 \, \text{m}
\]

Rounding to the nearest tenth:
\[
\boxed{1605.3}
\]

Problem 2:

The scale of a map is \( 2 \, \text{m} = 4 \, \text{mi} \).

- Map: \( 12 \, \text{m} \)
- Actual: \( \_\_\_\_\_ \, \text{mi} \)

#### Solution:
The scale factor is:
\[
\text{Scale factor} = \frac{4 \, \text{mi}}{2 \, \text{m}} = 2 \, \text{mi per m}
\]

For the map distance of \( 12 \, \text{m} \):
\[
\text{Actual distance} = 12 \, \text{m} \times 2 \, \text{mi per m} = 24 \, \text{mi}
\]

\[
\boxed{24}
\]

Problem 3:

The scale of a map is \( 2 \, \text{ft} = 10.4 \, \text{mi} \).

- Map: \( \_\_\_\_\_ \, \text{ft} \)
- Actual: \( 20.8 \, \text{mi} \)

#### Solution:
The scale factor is:
\[
\text{Scale factor} = \frac{10.4 \, \text{mi}}{2 \, \text{ft}} = 5.2 \, \text{mi per ft}
\]

For the actual distance of \( 20.8 \, \text{mi} \):
\[
\text{Map distance} = \frac{20.8 \, \text{mi}}{5.2 \, \text{mi per ft}} = 4 \, \text{ft}
\]

\[
\boxed{4}
\]

Problem 4:

The scale factor for a model is \( 7 \, \text{cm} = \_\_\_\_\_ \, \text{m} \).

- Model: \( 40.4 \, \text{cm} \)
- Actual: \( 80.6 \, \text{m} \)

#### Solution:
The scale factor is:
\[
\text{Scale factor} = \frac{80.6 \, \text{m}}{40.4 \, \text{cm}}
\]

Convert \( 40.4 \, \text{cm} \) to meters:
\[
40.4 \, \text{cm} = 0.404 \, \text{m}
\]

Now calculate:
\[
\text{Scale factor} = \frac{80.6}{0.404} \approx 200
\]

Thus, \( 7 \, \text{cm} \) corresponds to:
\[
7 \, \text{cm} \times 200 = 1400 \, \text{m}
\]

\[
\boxed{1400}
\]

Problem 5:

The scale of a map is \( 5 \, \text{m} = 15 \, \text{mi} \).

- Map: \( 7.8 \, \text{m} \)
- Actual: \( \_\_\_\_\_ \, \text{mi} \)

#### Solution:
The scale factor is:
\[
\text{Scale factor} = \frac{15 \, \text{mi}}{5 \, \text{m}} = 3 \, \text{mi per m}
\]

For the map distance of \( 7.8 \, \text{m} \):
\[
\text{Actual distance} = 7.8 \, \text{m} \times 3 \, \text{mi per m} = 23.4 \, \text{mi}
\]

\[
\boxed{23.4}
\]

Problem 6:

The scale of a map is \( 6 \, \text{km} = 4 \, \text{mi} \).

- Map: \( \_\_\_\_\_ \, \text{km} \)
- Actual: \( 40.1 \, \text{mi} \)

#### Solution:
The scale factor is:
\[
\text{Scale factor} = \frac{4 \, \text{mi}}{6 \, \text{km}} = \frac{2}{3} \, \text{mi per km}
\]

For the actual distance of \( 40.1 \, \text{mi} \):
\[
\text{Map distance} = \frac{40.1 \, \text{mi}}{\frac{2}{3} \, \text{mi per km}} = 40.1 \times \frac{3}{2} = 60.15 \, \text{km}
\]

\[
\boxed{60.2}
\]

Problem 7:

The scale factor for a model is \( 5 \, \text{cm} = \_\_\_\_\_ \, \text{m} \).

- Model: \( 40.2 \, \text{cm} \)
- Actual: \( 60.2 \, \text{m} \)

#### Solution:
The scale factor is:
\[
\text{Scale factor} = \frac{60.2 \, \text{m}}{40.2 \, \text{cm}}
\]

Convert \( 40.2 \, \text{cm} \) to meters:
\[
40.2 \, \text{cm} = 0.402 \, \text{m}
\]

Now calculate:
\[
\text{Scale factor} = \frac{60.2}{0.402} \approx 149.75
\]

Thus, \( 5 \, \text{cm} \) corresponds to:
\[
5 \, \text{cm} \times 149.75 = 748.75 \, \text{m}
\]

Rounding to the nearest tenth:
\[
\boxed{748.8}
\]

Problem 8:

The scale of a map is \( 4 \, \text{m} = 18 \, \text{mi} \).

- Map: \( 6.8 \, \text{m} \)
- Actual: \( \_\_\_\_\_ \, \text{mi} \)

#### Solution:
The scale factor is:
\[
\text{Scale factor} = \frac{18 \, \text{mi}}{4 \, \text{m}} = 4.5 \, \text{mi per m}
\]

For the map distance of \( 6.8 \, \text{m} \):
\[
\text{Actual distance} = 6.8 \, \text{m} \times 4.5 \, \text{mi per m} = 30.6 \, \text{mi}
\]

\[
\boxed{30.6}
\]

Problem 9:

The scale of a map is \( 4 \, \text{ft} = 16 \, \text{mi} \).

- Map: \( \_\_\_\_\_ \, \text{ft} \)
- Actual: \( 80 \, \text{mi} \)

#### Solution:
The scale factor is:
\[
\text{Scale factor} = \frac{16 \, \text{mi}}{4 \, \text{ft}} = 4 \, \text{mi per ft}
\]

For the actual distance of \( 80 \, \text{mi} \):
\[
\text{Map distance} = \frac{80 \, \text{mi}}{4 \, \text{mi per ft}} = 20 \, \text{ft}
\]

\[
\boxed{20}
\]

Problem 10:

The scale factor for a model is \( 8 \, \text{cm} = \_\_\_\_\_ \, \text{m} \).

- Model: \( 80.8 \, \text{cm} \)
- Actual: \( 90.6 \, \text{m} \)

#### Solution:
The scale factor is:
\[
\text{Scale factor} = \frac{90.6 \, \text{m}}{80.8 \, \text{cm}}
\]

Convert \( 80.8 \, \text{cm} \) to meters:
\[
80.8 \, \text{cm} = 0.808 \, \text{m}
\]

Now calculate:
\[
\text{Scale factor} = \frac{90.6}{0.808} \approx 112.13
\]

Thus, \( 8 \, \text{cm} \) corresponds to:
\[
8 \, \text{cm} \times 112.13 = 897.04 \, \text{m}
\]

Rounding to the nearest tenth:
\[
\boxed{897.0}
\]

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & 1605.3 \\
2. & 24 \\
3. & 4 \\
4. & 1400 \\
5. & 23.4 \\
6. & 60.2 \\
7. & 748.8 \\
8. & 30.6 \\
9. & 20 \\
10. & 897.0 \\
\end{array}
}
\]