Rukasha Maths

GCSE maths Beta

Direct and inverse proportion equations · Revision guide

Higher · Ratio · 1 page in the PDF

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Learning objectives

  • Write direct or inverse proportion as an equation.
  • Find the constant from a known pair and use it for a new input.

Key ideas

Direct proportion to a positive integer power includes that power in the model.

y∝xn⇒y=kxn,n∈Z>0y\propto x^n\quad\Rightarrow\quad y=kx^n,\quad n\in\mathbb{Z}_{>0}

Inverse proportion to a positive integer power places that power in the denominator.

y=kxn,x≠0,n∈Z>0y=\frac{k}{x^n},\quad x\ne0,\quad n\in\mathbb{Z}_{>0}

Use a known pair to find the constant before substituting a new input.

For positive inputs in this inverse-root model, quadrupling the input halves the output.

y=kx,x>0y=\frac{k}{\sqrt x},\qquad x>0

Worked examples

Example 1

The output is directly proportional to the square of the input. The known pair is three and eighteen. Find the output at five.

y=kx2,x=3, y=18⇒18=9ky=kx^2,\quad x=3,\ y=18\quad\Rightarrow\quad18=9k
k=2⇒y=2x2k=2\quad\Rightarrow\quad y=2x^2
x=5⇒y=2(52)x=5\quad\Rightarrow\quad y=2(5^2)
y=50y=50

Example 2

For inverse proportion, use the known pair two and twelve to find the output at six.

y=kx,x=2, y=12⇒k=24y=\frac{k}{x},\quad x=2,\ y=12\quad\Rightarrow\quad k=24
y=246=4y=\frac{24}{6}=4

Watch out

Do not omit the specified power or root. Doubling an input does not always double its output.