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Before reading this guide, it may be helpful to read the guide from Module 7 (M7) about proportion and variation.

Module 7 (M7) proportion and variation is concerned with direct proportion between two variables.

There are 3 types of direct proportion to be considered:

• \( p \propto q \)

• \( p \propto q^{2}\)

• \( p \propto \sqrt{q}\)

For all three, as \(q\) increase, \(p\) also increases.

For inverse proportion, one quantity decreases as the other increases. For example, if speed is increased, the time taken for a journey will decrease.

\(t\) is inversely proportional to \(v\) can be written as \(t \propto \frac{1}{v}\)

Inverse proportion is also referred to as indirect proportion.

The rules for setting up an equation are the same as for direct proportion.

1. Write down how the variables are connected using the proportion sign, \( \propto\)
2. Rewrite using \(k\) and an equal sign (\(=\))
3. Substitute the values given for the two variables into this equation.
4. Solve to find a value for \(k\).
5. Rewrite the equation using this value for \(k\).
6. Use the equation as a formula to find any unknown values.

Example

\(g\) is inversely proportional to \(f\). When \(f = 0.4\), \(g = 13\). Find a formula connecting \(g\) and \(f\) and use it to find the value of \(f\) when \(g = 20\).

Solution

• Write down how the variables are connected, using the proportion sign, \(\propto\)

\(g \propto \frac{1}{f}\)

• Rewrite using \(k\) and an equal sign (\(=\))

\(g = \frac{k}{f}\)

• Substitute the values given for the two variables into this equation.

When \(f = 0.4\), \(g = 20\)

\(13 = \frac{k}{0.4}\)

• Solve to find a value for \(k\).

\(k \times 0.4 = 5.2\)

• Rewrite the equation using this value for \(k\).

\(g = \frac{5.2}{f}\)

• Use the equation as a formula to find any unknown values.

If \(g = 20\), \(f = ?\)

\(20 = \frac{5.2}{f}\)

\(f = \frac{5.2}{20} = 0.26\)

Answer

\(f = 0.26\)