# Inverse proportion

Two variables $$\Tblue{a}$$ and $$\Tgreen{b}$$ are inversely proportional if $$\Tblue{a}$$ is directly proportional to the **reciprocal** of $$\Tgreen{b}$$. We write $$\Tblue{a}\propto 1/\Tgreen{b}$$.

To cover a given distance, the time taken is inversely proportional to speed. The higher the speed, the shorter the time.

If $$\Tblue{a}$$ and $$\Tgreen{b}$$ are inversely proportional, their product is a constant $$\Tred{k}$$. $$\Tblue{a}\Tgreen{b}=\Tred{k},\qquad \Tblue{a} = \frac{\Tred{k}}{\Tgreen{b}}.$$

Several variables are collected in the table below. The variables $$\Tblue{a}$$ and $$\Tgreen{b}$$ are proportional with proportionality constant $$\Tred{10}$$. The variables $$\Tblue{a}$$ and $$\Tgreen{c}$$ are not proportional.

Variable | $$\Tblue{a}$$ | $$\Tblue{1}$$ | $$\Tblue{2}$$ | $$\Tblue{3}$$ | $$\Tblue{4}$$ | $$\Tblue{5}$$ |
---|---|---|---|---|---|---|

Variable | $$\Tgreen{b}$$ | $$\Tgreen{10}$$ | $$\Tgreen{5}$$ | $$\Tgreen{3.33}$$ | $$\Tgreen{2.5}$$ | $$\Tgreen{2}$$ |

Product | $$\Tred{a\times b}$$ | $$\Tred{10}$$ | $$\Tred{10}$$ | $$\Tred{10}$$ | $$\Tred{10}$$ | $$\Tred{10}$$ |

Variable | $$\Tgreen{c}$$ | $$\Tgreen{10}$$ | $$\Tgreen{25}$$ | $$\Tgreen{11.11}$$ | $$\Tgreen{6.25}$$ | $$\Tgreen{4}$$ |

Product | $$\Tred{a\times c}$$ | $$\Tred{10}$$ | $$\Tred{50}$$ | $$\Tred{33.33}$$ | $$\Tred{25}$$ | $$\Tred{20}$$ |

From the table, we can deduce that the proportionality constant is $$\Tred{10}$$.