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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 $$\Tred{a/c}$$ $$\Tred{1}$$ $$\Tred{0.5}$$ $$\Tred{0.33}$$ $$\Tred{0.25}$$ $$\Tred{0.2}$$
The price paid is directly proportional to the weight of the fruit. The proportionality constant depends on the units used to weigh the fruit.
The price paid is directly proportional to the weight of the fruit. The proportionality constant depends on the units used to weigh the fruit.

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 $$\Tred{10}$$.