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# Proportion

## Direct 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 Variable Ratio Variable Ratio $\Tblue{a}$ $\Tblue{3}$ $\Tblue{6}$ $\Tblue{9}$ $\Tblue{12}$ $\Tblue{15}$ $\Tgreen{b}$ $\Tgreen{1}$ $\Tgreen{2}$ $\Tgreen{3}$ $\Tgreen{4}$ $\Tgreen{5}$ $\Tred{a/b}$ $\Tred{3}$ $\Tred{3}$ $\Tred{3}$ $\Tred{3}$ $\Tred{3}$ $\Tgreen{c}$ $\Tgreen{3}$ $\Tgreen{12}$ $\Tgreen{27}$ $\Tgreen{48}$ $\Tgreen{75}$ $\Tred{a/c}$ $\Tred{1}$ $\Tred{0.5}$ $\Tred{0.33}$ $\Tred{0.25}$ $\Tred{0.2}$
The price paid is directly proportional to the weight of the fruit. The proportionality constant depends on the units used to weigh the fruit.

## 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 Variable Product Variable Product $\Tblue{a}$ $\Tblue{1}$ $\Tblue{2}$ $\Tblue{3}$ $\Tblue{4}$ $\Tblue{5}$ $\Tgreen{b}$ $\Tgreen{10}$ $\Tgreen{5}$ $\Tgreen{3.33}$ $\Tgreen{2.5}$ $\Tgreen{2}$ $\Tred{a\times b}$ $\Tred{10}$ $\Tred{10}$ $\Tred{10}$ $\Tred{10}$ $\Tred{10}$ $\Tgreen{c}$ $\Tgreen{10}$ $\Tgreen{25}$ $\Tgreen{11.11}$ $\Tgreen{6.25}$ $\Tgreen{4}$ $\Tred{a\times c}$ $\Tred{10}$ $\Tred{50}$ $\Tred{33.33}$ $\Tred{25}$ $\Tred{20}$

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