Ratios and Proportions • Topic 6 of 7

Scale Factors

A scale factor is the ratio of a length in a copy to the matching length in the original. A scale factor greater than 1 enlarges; between 0 and 1 it reduces. Lengths multiply by the scale factor k, areas by k², and volumes by k³ — a fact the SAT loves to test. On maps and blueprints the scale (for example 1 : 50) tells you how many real-world units one drawing unit represents. To find a missing real length, multiply the drawing length by the scale; to find a drawing length, divide. Keep units consistent throughout.

✅ Solved examples

1. A drawing uses a scale of 1 : 100. A wall is 4 cm on the drawing. How long is it in real life?
Multiply by 100: 4 × 100 = 400 cm = 4 m.
2. A photo is enlarged by a scale factor of 3. A 5 cm length becomes how long?
5 × 3 = 15 cm.
3. If a shape is scaled by factor 2, how does its area change?
Area scales by k² = 2² = 4, so the area is 4 times larger.
4. A model car is 1/20 the real size. The model is 25 cm long. How long is the real car?
Scale factor 20: 25 × 20 = 500 cm = 5 m.

✏️ Practice — try these, take hints as needed

1. A map scale is 1 : 200. A road is 6 cm on the map. What is its real length?
Multiply the map length by 200.
6 × 200 = 1200 cm.
Convert to metres if needed.
1200 cm (12 m).
2. A figure is scaled by factor 4. How does its area change?
Area scales by the square of the factor.
4² = 16.
State the multiple.
16 times larger.
3. A 3 cm line is enlarged by scale factor 5. What is the new length?
Multiply by the scale factor.
3 × 5.
Compute.
15 cm.
4. A cube is scaled by factor 2. How does its volume change?
Volume scales by the cube of the factor.
2³ = 8.
State the multiple.
8 times larger.
5. A model is 1/50 of real size and is 8 cm long. What is the real length?
Scale factor is 50.
8 × 50.
Convert to metres.
400 cm (4 m).

📝 Topic test — 8 questions

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