Scaling Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Scaling.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Changing the size of a quantity by multiplying by a factor, making it proportionally larger (factor > 1) or smaller (factor < 1).

Zooming in or out—everything gets bigger or smaller by the same factor.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Scaling preserves all proportions while changing overall size—ratios between parts stay the same.

Common stuck point: Area scales by k^2 and volume by k^3—doubling lengths quadruples area and octuples volume.

Sense of Study hint: Write out every quantity in the problem, then multiply each one by the same scale factor. Check that nothing got skipped.

Worked Examples

Example 1

easy
A map has a scale of 1:25{,}000. Two cities are 8 cm apart on the map. What is the actual distance in kilometres?

Solution

  1. 1
    The scale 1:25{,}000 means 1 cm on the map represents 25{,}000 cm in reality.
  2. 2
    Actual distance = 8 \times 25{,}000 = 200{,}000 cm.
  3. 3
    Convert to kilometres: 200{,}000 \text{ cm} \div 100{,}000 = 2 km.

Answer

The actual distance is 2 km.
A map scale is a ratio expressing how much the real world has been shrunk. Multiplying the map measurement by the scale ratio gives the real-world measurement in the same units, which can then be converted as needed.

Example 2

medium
A recipe for 4 servings uses 2.5 cups of oats, 1.5 cups of milk, and \frac{1}{4} cup of honey. Scale the recipe up to 10 servings.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A model car is built at a scale of 1:18. The model is 24 cm long. How long is the actual car in metres?

Example 2

medium
A square has side length 5 cm. It is scaled by a factor of 3. By what factor does the area increase?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

multiplicationratios