# Direct and inverse proportion

## Direct proportion

There is a direct proportion between two values when one is a multiple of the other. For example, . 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 .

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

If then is proportional to and can be calculated for :

Similarly, if then can be calculated:

To find , divide 60 by 2:

## 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 is directly proportional to . When , . Find an equation relating and .

1. so

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

If then, .

## Inverse proportion

If one value is inversely proportional to another then it is written using the proportionality symbol 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:

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

Using: (so is inversely proportional to ).

If then find .

Similarly, if , find .

## 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 is inversely proportional to w and when , , then form an equation relating to .

1. so

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

If then find .

Similarly, if , find .