Free Collection of Scale Factor Worksheets | WorksheetZone - Free Printable

Let's solve each problem step by step using scale factors. The key idea is that:

> Scale factor = (Model dimension) / (Actual dimension)
> or
> (Model) / (Actual) = (Given scale)

We will use proportions to find missing values.

---

Problem 1

The scale factor for a model is 5 cm = _____ m
Model: 9.5 cm
Actual: 30.5 m

We are given the model and actual dimensions, so we can find the scale in meters.

Set up a proportion:
$$
\frac{5 \text{ cm}}{x \text{ m}} = \frac{9.5 \text{ cm}}{30.5 \text{ m}}
$$

Cross-multiply:
$$
5 \times 30.5 = 9.5 \times x \\
152.5 = 9.5x \\
x = \frac{152.5}{9.5} \approx 16.05 \text{ m}
$$

But wait — this is not what we're asked. The question says:

> "The scale factor for a model is 5 cm = _____ m"

So we need to find how many meters correspond to 5 cm, based on the ratio of model to actual.

From the given:
- Model: 9.5 cm → Actual: 30.5 m

So, 9.5 cm corresponds to 30.5 m
Then, 5 cm corresponds to?

Use proportion:
$$
\frac{9.5}{30.5} = \frac{5}{x} \\
9.5x = 5 \times 30.5 = 152.5 \\
x = \frac{152.5}{9.5} \approx 16.05 \text{ m}
$$

Wait — that would mean 5 cm = 16.05 m? That seems large.

But let’s double-check: If 9.5 cm represents 30.5 m, then:

$$
\text{Scale factor} = \frac{9.5}{30.5} \text{ cm/m} \Rightarrow \text{so } 1 \text{ m} = \frac{9.5}{30.5} \text{ cm}
$$

But we want to know: How many meters does 5 cm represent?

So:
$$
\frac{9.5 \text{ cm}}{30.5 \text{ m}} = \frac{5 \text{ cm}}{x \text{ m}} \\
x = \frac{5 \times 30.5}{9.5} = \frac{152.5}{9.5} \approx 16.05 \text{ m}
$$

So, 5 cm = 16.05 m, rounded to nearest tenth: 16.1 m

✔ Answer: 16.1 m

---

Problem 2

The scale of a map is 2 m = 4 mi
Map: 12 m
Actual: _____ mi

Given: 2 m on map = 4 miles actual

So, scale: 2 m : 4 mi → simplify: 1 m : 2 mi

So, 12 m on map = ?

$$
\frac{2}{4} = \frac{12}{x} \Rightarrow 2x = 48 \Rightarrow x = 24 \text{ mi}
$$

✔ Answer: 24 mi

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Problem 3

The scale of a map is 2 ft = 10.4 mi
Map: _____ ft
Actual: 20.8 mi

Set up proportion:
$$
\frac{2}{10.4} = \frac{x}{20.8} \\
2 \times 20.8 = 10.4x \\
41.6 = 10.4x \\
x = \frac{41.6}{10.4} = 4 \text{ ft}
$$

✔ Answer: 4 ft

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Problem 4

The scale factor for a model is 7 cm = _____ m
Model: 40.4 cm
Actual: 80.6 m

Find how many meters correspond to 7 cm.

Use proportion:
$$
\frac{7}{x} = \frac{40.4}{80.6} \\
7 \times 80.6 = 40.4 \times x \\
564.2 = 40.4x \\
x = \frac{564.2}{40.4} \approx 13.96 \text{ m}
$$

Rounded to nearest tenth: 14.0 m

✔ Answer: 14.0 m

---

Problem 5

The scale of a map is 5 m = 15 mi
Map: 7.8 m
Actual: _____ mi

Scale: 5 m → 15 mi → 1 m → 3 mi

So, 7.8 m → 7.8 × 3 = 23.4 mi

Or via proportion:
$$
\frac{5}{15} = \frac{7.8}{x} \Rightarrow 5x = 117 \Rightarrow x = 23.4
$$

✔ Answer: 23.4 mi

---

Problem 6

The scale of a map is 6 km = 4 mi
Map: _____ km
Actual: 40.1 mi

Set up proportion:
$$
\frac{6}{4} = \frac{x}{40.1} \\
6 \times 40.1 = 4x \\
240.6 = 4x \\
x = 60.15 \text{ km}
$$

Rounded to nearest tenth: 60.2 km

✔ Answer: 60.2 km

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Problem 7

The scale factor for a model is 5 cm = _____ m
Model: 40.2 cm
Actual: 60.2 m

Use proportion:
$$
\frac{5}{x} = \frac{40.2}{60.2} \\
5 \times 60.2 = 40.2x \\
301 = 40.2x \\
x = \frac{301}{40.2} \approx 7.487 \text{ m}
$$

Rounded to nearest tenth: 7.5 m

✔ Answer: 7.5 m

---

Problem 8

The scale of a map is 4 m = 18 mi
Map: 6.8 m
Actual: _____ mi

Proportion:
$$
\frac{4}{18} = \frac{6.8}{x} \\
4x = 18 \times 6.8 = 122.4 \\
x = \frac{122.4}{4} = 30.6 \text{ mi}
$$

✔ Answer: 30.6 mi

---

Problem 9

The scale of a map is 4 ft = 16 mi
Map: _____ ft
Actual: 80 mi

Scale: 4 ft → 16 mi → 1 ft → 4 mi

So, 80 mi ÷ 4 = 20 ft

Or:
$$
\frac{4}{16} = \frac{x}{80} \\
4 \times 80 = 16x \\
320 = 16x \\
x = 20 \text{ ft}
$$

✔ Answer: 20 ft

---

Problem 10

The scale factor for a model is 8 cm = _____ m
Model: 80.8 cm
Actual: 90.6 m

Use proportion:
$$
\frac{8}{x} = \frac{80.8}{90.6} \\
8 \times 90.6 = 80.8x \\
724.8 = 80.8x \\
x = \frac{724.8}{80.8} \approx 9.0 \text{ m}
$$

✔ Answer: 9.0 m

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | 16.1 m |
| 2 | 24 mi |
| 3 | 4 ft |
| 4 | 14.0 m |
| 5 | 23.4 mi |
| 6 | 60.2 km |
| 7 | 7.5 m |
| 8 | 30.6 mi |
| 9 | 20 ft |
| 10 | 9.0 m |

All answers are rounded to the nearest tenth as requested.

Let me know if you'd like this formatted as a completed worksheet!