Practice Scaling Laws in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Relationships describing how a quantity changes when the size or scale of a system is multiplied by a factor, often expressed as power laws.

When you double the length of a cube, its volume grows by 23=8. Scaling laws reveal how fast quantities grow — they often explain why small and large things behave so differently.

Showing a random 20 of 50 problems.

Example 1

easy
A cube has volume 8 cm3. If the edge is tripled, find the new volume.

Example 2

easy
A recipe's cost is proportional to the number of servings. If servings triple, what happens to cost?

Example 3

medium
Two similar cones have volumes 8 cm3 and 125 cm3. Find the ratio of their heights.

Example 4

medium
A scale model car has volume 0.001 m3 at scale 1:20. Find the actual volume.

Example 5

easy
If y∝x3 and x is tripled, by what factor does y change?

Example 6

easy
If a cube's edge is tripled, by what factor does its volume change?

Example 7

medium
A sphere of radius r has volume V=43πr3. If the radius doubles, by what factor does the volume increase? Generalise to a scaling factor k.

Example 8

medium
A cube's surface-area-to-volume ratio is 6/s for edge s. If the edge is multiplied by 4, what happens to this ratio?

Example 9

easy
Two similar triangles have a side ratio of 1:4. What is the ratio of their areas?

Example 10

medium
A quantity follows y = 5 x^3. If x increases by 20% (factor 1.2), by what percent does y increase?

Example 11

easy
If a triangle's three sides are all halved, by what factor does its area change?

Example 12

hard
A 6-inch pizza costs $8 and a 12-inch pizza costs $24. Which is the better deal per square inch?

Example 13

easy
A photo is enlarged so each side triples. By what factor does its area increase?

Example 14

easy
If the radius of a circle doubles, by what factor does the circumference change? What about the area?

Example 15

medium
A balloon's volume scales as r^3 and is being inflated so r grows steadily. When r has doubled, the surface area (~r^2) has grown by what factor compared to volume's growth?

Example 16

medium
If a sphere has surface area S and radius r is multiplied by 5, find the new surface area in terms of S.

Example 17

easy
If you double the edge of a cube, by what factor does its volume change?

Example 18

easy
Two similar rectangles have a length ratio of 1:5. What is the ratio of their areas?

Example 19

hard
If y∝x−2 (inverse square law) and x is doubled, by what factor does y change?

Example 20

easy
If a square's side is tripled, by what factor does its area change?
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