# Direct and inverse proportion

## Direct proportion

There is a direct proportion between two values when one is a multiple of the other. For example, $$1 \:\text{cm} = 10 \:\text{mm}$$. To convert cm to mm, the is always 10. Direct proportion is used to calculate the cost of petrol or exchange rates of foreign money.

The symbol for direct proportion is $$\propto$$.

The statement ‘t is directly proportional to r’ can be written using the symbol:

$t \propto r$

If $$y = 2p$$ then $$y$$ is proportional to $$p$$ and $$y$$ can be calculated for $$p = 7$$:

$y = 2 \times 7 = 14$

Similarly, if $$y = 60$$ then $$p$$ can be calculated:

$60 = 2p$

To find $$p$$, divide 60 by 2:

$60 \div 2 = 30$

## Direct proportion - Higher

Proportionality can be used to set up an .

There are four steps to do this:

1. write the proportional relationship
2. convert to an equation using a constant of proportionality
3. use given information to find the constant of proportionality
4. substitute the constant of proportionality into the equation

### Example

The value $$e$$ is directly proportional to $$p$$. When $$e = 20$$, $$p = 10$$. Find an equation relating $$e$$ and $$p$$.

1. $e \propto p$
2. $e = kp$
3. $$20 = 10k$$ so $$k = 20 \div 10 = 2$$
4. $e = 2p$

This equation can now be used to calculate other values of $$e$$ and $$p$$.

If $$p = 6$$ then, $$e = 2 \times 6 = 12$$.

## Inverse proportion

If one value is inversely proportional to another then it is written using the proportionality symbol $$\propto$$ in a different way. Inverse proportion occurs when one value increases and the other decreases. For example, more workers on a job would reduce the time to complete the task. They are inversely proportional.

The statement ‘b is inversely proportional to m’ is written:

$b \propto \frac{1}{m}$

Equations involving inverse proportions can be used to calculate other values.

Using: $$g = \frac{36}{w}$$ (so $$g$$ is inversely proportional to $$w$$).

If $$g = 8$$ then find $$w$$.

$8 = \frac{36}{w}$

$w = \frac{36}{8} = 4.5$

Similarly, if $$w = 6$$, find $$g$$.

$g = \frac{36}{6}$

$g = 6$

## Inverse proportion - Higher

Proportionality can be used to set up an equation.

There are four steps to do this:

1. write the proportional relationship
2. convert to an equation using a constant of proportionality
3. use given information to find the constant of proportionality
4. substitute the constant of proportionality into the equation

### Example

If $$g$$ is inversely proportional to w and when $$g = 4$$, $$w = 9$$, then form an equation relating $$g$$ to $$w$$.

1. $g \propto \frac{1}{w}$
2. $g = k \times \frac{1}{w} = \frac{k}{w}$
3. $$4 = \frac{k}{9}$$ so $$k = 4 \times 9 = 36$$
4. $g = \frac{36}{w}$

This equation can be used to calculate new values of $$g$$ and $$w$$.

If $$g = 8$$ then find $$w$$.

$8 = \frac{36}{w}$

$w = \frac{36}{8} = 4.5$

Similarly, if $$w = 6$$, find $$g$$.

$g = \frac{36}{6}$

$g = 6$