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 \(\propto\).

\(t \propto r\)

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:

\(60 \div 2 = 30\)

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, \(e = 2 \times 6 = 12\).

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.

\(b \propto \frac{1}{m}\)

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 \(g\).

\(g = \frac{36}{6}\)

\(g = 6\)

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 \(g\).

\(g = \frac{36}{6}\)

\(g = 6\)