5.4 Inverse proportion — Learn

Prerequisites

You should be able to multiply and divide whole numbers and fractions. Check units before comparing quantities.

Key terms

Inverse proportion means the product of two quantities stays fixed. The constant of proportionality is written $k$.

Objectives

• Find equations and values in inverse and inverse-square proportion.
• Check that the result matches the units and direction of change.

Learn the method

Inverse proportion means the product stays fixed. For $y\propto1/x$, write $y=k/x$, so $xy=k$. For inverse square proportion, write $y=k/x^2$. The Higher extension asks you to construct an equation from a known pair of values.

Worked example. 4 workers take 18 days. How long would 6 workers take?

• Find the fixed product: $4\times18=72$ worker-days.
• Divide by 6 workers: $72\div6=12$ days.
• Check: $6\times12=72$ worker-days.

Worked example. A graph follows $y=24/x$. Find $y$ when $x=6$, and describe what happens when $x$ doubles.

• Substitute $x=6$: $y=24\div6=4$.
• If $x$ doubles, the denominator doubles, so $y$ halves.
• Check: when $x=12$, $y=24\div12=2$.

Higher only. If $y$ is inversely proportional to $x$, write $y=k/x$. Use a known pair to find $k$, then substitute a new value.

Watch out. In inverse proportion, doubling $x$ halves $y$. Do not use the direct-proportion rule; check that $xy$ stays constant.

You should now be able to

• I can find $k$ in an inverse or inverse-square relationship.
• I can use a fixed product to find a missing value.
• I can check that the product stays constant.