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MathidiaMathematics Academy
Proportion and rates of change
Lesson 42 of 75
Easy practice

Inverse proportion with powers

Objective. Use y=k/xⁿ models in unfamiliar contexts.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Inverse proportion with powers means y=k/xⁿ, so y decreases as a power of x increases, rather than as x alone. To find k, substitute a known pair of values into y=k/xⁿ and solve, then use the formula for other values. If y is inversely proportional to x² and y=4 when x=5, then 4=k/25, so k=100, giving y=100/x². This means tripling x in y=k/x² reduces y to a ninth of its value, since (3x)²=9x² makes the denominator 9 times larger. Always read the wording carefully to identify the correct power n before setting up the equation.

Worked example

y is inversely proportional to the square of x. When x = 2, y = 20. Find y when x = 4.

Diagram of a square
  1. 1Write y = k/x², substitute x=2, y=20: 20 = k/4
  2. 2Solve for k: k = 80
  3. 3Formula is y = 80/x²
  4. 4Substitute x=4: y = 80/16 = 5

Answer: 5

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Common mistake. Pupils use y=kxⁿ instead of y=k/xⁿ for inverse proportion, treating it as direct proportion by mistake. This gives y increasing with x rather than decreasing. Always check whether the relationship is "proportional to" (direct) or "inversely proportional to" (divide by the power).

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