# Proportion

Two variables $$\Tblue{a}$$ and $$\Tgreen{b}$$ are proportional, or in direct proportion, if the ratio $$\Tblue{a}/\Tgreen{b}$$ is constant for all values of the variables.

At constant speed, distance covered is proportional to time.

We write $$\Tblue{a}\,\Torange{\propto}\,\Tgreen{b}$$ using the proportionality symbol $$\Torange{\propto}$$.

The ratio $$\Tred{k}$$ between proportional variables is the proportionality constant. $$\Tblue{a}=\Tred{k}\,\Tgreen{b},\qquad \Tblue{a}:\Tgreen{b} = 1:\Tred{k}.$$

Several variables are collected in the table below. The variables $$\Tblue{a}$$ and $$\Tgreen{b}$$ are proportional with proportionality constant $$\Tred{3}$$. The variables $$\Tblue{a}$$ and $$\Tgreen{c}$$ are not proportional.

Variable | $$\Tblue{a}$$ | $$\Tblue{3}$$ | $$\Tblue{6}$$ | $$\Tblue{9}$$ | $$\Tblue{12}$$ | $$\Tblue{15}$$ |
---|---|---|---|---|---|---|

Variable | $$\Tgreen{b}$$ | $$\Tgreen{1}$$ | $$\Tgreen{2}$$ | $$\Tgreen{3}$$ | $$\Tgreen{4}$$ | $$\Tgreen{5}$$ |

Ratio | $$\Tred{a/b}$$ | $$\Tred{3}$$ | $$\Tred{3}$$ | $$\Tred{3}$$ | $$\Tred{3}$$ | $$\Tred{3}$$ |

Variable | $$\Tgreen{c}$$ | $$\Tgreen{3}$$ | $$\Tgreen{12}$$ | $$\Tgreen{27}$$ | $$\Tgreen{48}$$ | $$\Tgreen{75}$$ |

Ratio | $$\Tred{a/c}$$ | $$\Tred{1}$$ | $$\Tred{0.5}$$ | $$\Tred{0.33}$$ | $$\Tred{0.25}$$ | $$\Tred{0.2}$$ |

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 $$\Tred{10}$$.