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 multiplier 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 proportionality 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 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

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 \(g\).

\[g = \frac{36}{6}\]

\[g = 6\]