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      Finding the constant of proportionality for inverse proportion

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      Lesson details

      Learning outcome

      I can use the general form for an inversely proportional relationship to find k.

      Key learning points

      1. Inverse proportion equations are of the form y = k ÷ x^n
      2. The equation can be written in the form yx^n = k
      3. k is the constant of proportionality.
      4. To find k, you can use a pair of values.

      Keywords

      • Inverse proportion - Two variables are inversely proportional if there is a constant multiplicative relationship between one variable and the reciprocal of the other

      Common misconception

      For all direct and inverse proportions, the multiplicative relationship is with x and y.

      The multiplicative relationship whether direct or inverse can be shown as y and x^n. Show a ratio table of y and 1/x^2 and a table of y and 1/x. The inverse multiplicative relationship is seen with y and 1/x^2 and not with 1/x.

      Teacher tip

      Get pupils to invent their own inverse proportion equation. From this equation, create their own questions so to find one variable given another variable. These questions can be shared with peers. For a greater challenge, there are opportunities to use fractions, surds & a non integer exponents.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
       except where otherwise stated. See Oak's terms & conditions
       (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      Download starter quiz questions (PDF)

      6 Questions

      Q1.
      Match the statement with the correct notation.

      Correct Answer:$$y\propto x$$,$$y$$ is proportional to $$x$$

      $$y$$ is proportional to $$x$$

      Correct Answer:$$y\propto x^2$$,$$y$$ is proportional to $$x^2$$

      $$y$$ is proportional to $$x^2$$

      Correct Answer:$$y \propto \frac{1}{x^2}$$,$$y$$ is inversely proportional to $$x^2$$

      $$y$$ is inversely proportional to $$x^2$$

      Correct Answer:$$y \propto \frac{1}{x}$$,$$y$$ is inversely proportional to $$x$$

      $$y$$ is inversely proportional to $$x$$

      Q2.
      Match the coordinates with the directly proportional equations.

      Correct Answer:(4,12) ,$$y=3x$$

      $$y=3x$$

      Correct Answer:(9,3) ,$$y=\frac {x}{3}$$

      $$y=\frac {x}{3}$$

      Correct Answer:(6,8) ,$$y=\frac {4x}{3}$$

      $$y=\frac {4x}{3}$$

      Correct Answer:(10,2) ,$$y=\frac {x}{5}$$

      $$y=\frac {x}{5}$$

      Correct Answer:(2,10) ,$$y=5x$$

      $$y=5x$$

      Q3.
      $$y$$ is proportional to $$x$$. When $$y$$ = 48, $$x$$ = 12. What is the equation to show the direct proportion between $$x$$ and $$y$$?

      $$y=2x$$
      $$y=\frac{4}{x}$$
      Correct answer: $$y=4x$$
      $$y=\frac{2}{x}$$

      Q4.
      $$y$$ is proportional to $$x$$. When $$y$$ = 20, $$x$$ = 80. What is the equation to show the direct proportion between $$x$$ and $$y$$?

      $$y=\frac{1}{4x}$$
      $$y=\frac{0.25}{x}$$
      $$y=\frac{4}{x}$$
      Correct answer: $$y=\frac{x}{4}$$

      Q5.
      $$y$$ is proportional to $$x^2$$. When $$y$$ = 200, $$x$$ = 10. What is the equation to show the direct proportion between $$x$$ and $$y$$?

      $$y=2x$$
      $$y=(2x)^2$$
      Correct answer: $$y=2x^2$$
      $$y=\frac{x^2}{2}$$

      Q6.
      $$y$$ is proportional to $$\sqrt{x}$$. When $$y$$ = 2, $$x$$ = 36. What is the equation to show the direct proportion between $$x$$ and $$y$$?

      $$y=\frac{1}{\sqrt{x}}$$
      $$y=\frac{4}{\sqrt{x}}$$
      Correct answer: $$y=\frac{\sqrt{x}}{3}$$
      $$y=\frac{3}{\sqrt{x}}$$

      Assessment exit quiz

      Download exit quiz questions (PDF)

      6 Questions

      Q1.
      Match the constant of proportionality for each of the following inverse proportions.

      Correct Answer:$$y=\frac{2}{x}$$,2

      2

      Correct Answer:$$y=\frac{1}{2x}$$,$$\frac{1}{2}$$

      $$\frac{1}{2}$$

      Correct Answer:$$y=\frac{3}{2x}$$,1.5

      1.5

      Correct Answer:$$y=\frac{2}{3x}$$,$$\frac{2}{3}$$

      $$\frac{2}{3}$$

      Q2.
      Match the correct proportionality statement with an example equation.

      Correct Answer:$$y=4x$$,$$y$$ is proportional to $$x$$

      $$y$$ is proportional to $$x$$

      Correct Answer:$$y=4x^2$$,$$y$$ is proportional to $$x^2$$

      $$y$$ is proportional to $$x^2$$

      Correct Answer:$$y = \frac{4}{x^2}$$,$$y$$ is inversely proportional to $$x^2$$

      $$y$$ is inversely proportional to $$x^2$$

      Correct Answer:$$y =\frac{4}{x}$$,$$y$$ is inversely proportional to $$x$$

      $$y$$ is inversely proportional to $$x$$

      Q3.
      $$y$$ is $$\propto \frac{1}{x}$$. When $$y$$ = 4, $$x = 2$$. What is the equation to show the inversely proportional relationship between $$x$$ and $$y$$?

      $$y=8x$$
      $$y=\frac{8}{3x}$$
      $$y=\frac{x}{8}$$
      Correct answer: $$y=\frac{8}{x}$$
      $$y=\frac{24}{x}$$

      Q4.
      $$y$$ is $$\propto \frac{1}{x}$$. When $$y=\frac{1}{5}$$, $$x=2$$. Work out $$y$$ when $$x=10$$. Give your answer as a decimal.

      Correct Answer: 0.04

      Q5.
      $$y$$ is $$\propto \frac{1}{x^2}$$. When $$y=\frac{1}{6}$$, $$x=2$$. Work out $$y$$ when $$x=\sqrt{8}$$.

      12
      Correct answer: $$\frac{1}{12}$$
      $$\frac{2}{3}$$
      $$\frac{3}{2}$$

      Q6.
      $$y$$ is $$\propto \frac{1}{\sqrt{x}}$$. When $$y=2.4$$, $$x=100$$. Work out $$y$$ when $$x=64$$.

      Correct Answer: 3

      To help you plan your 11 maths lesson on: Finding the constant of proportionality for inverse proportion, download all teaching resources for free and adapt to suit your pupils' needs...