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.
Inverse proportion
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.
- Write down how the variables are connected using the proportion sign, \( \propto\)
- Rewrite using \(k\) and an equal sign (\(=\))
- Substitute the values given for the two variables into this equation.
- Solve to find a value for \(k\).
- Rewrite the equation using this value for \(k\).
- 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
\(f = 0.26\)
Question
The force of attraction (\(F\)) between two identical magnets is inversely proportional to the square of the distance (\(d\)) between them. When the magnets are 5 cm apart, the force is found to be 16 newtons. How far apart are the magnets when the force is 20 newtons?
Give an exact answer.
Answer
\(F \propto \frac{1}{d^{2}}\)
\(F = \frac{k}{d^{2}}\)
\(F = 16\) when \(d = 5\), therefore
\(16 = \frac{k}{5^{2}}\)
\(k = 16 \times 25 = 400\)
\(F = \frac{400}{d^{2}}\)
\(F = 20\) when \(d = ?\)
\(20 = \frac{400}{d^{2}}\)
\(20d^{2} = 400\)
\(d = \sqrt{20}\)
Answer
\(d = \sqrt{20}\) or \(2\sqrt{5}\)
Keep the square root for an exact answer!
Question
\(W\) varies inversely as the square root of \(z\).
When \(z = 0.36\), \(W = 200\)
Find the value of \(W\) when \(z = 0.64\).
Answer
\(W\) varies inversely as the square root of \(z\).
\(W \propto \frac{1}{\sqrt{z}}\)
\(W =\frac{k}{\sqrt{z}}\)
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