Direct and inverse proportion questions are about how two quantities change in relation to each other.
The key is to recognise whether the two quantities increase together or whether one increases while the other decreases.
Direct proportion
Both quantities increase or decrease together.
For example, if one chocolate bar costs £2:
- 1 bar costs £2.
- 2 bars cost £4.
- 3 bars cost £6.
- 4 bars cost £8.
If you double the number of bars, you double the cost.
The formula for direct proportion is:
\[ y=kx \]
Here, $k$ is the constant of proportionality.
Worked example 1
$y$ is directly proportional to $x$.
When $x=3$, $y=12$. Find $y$ when$x=5$.
Step 1: Write the formula.
\[ y=kx \]
Step 2: Find $k$.
Substitute $x=3$ and $y=12$:
\[ 12=3k \]
\[ k=12\div3=4 \]
Step 3: Write the equation.
\[ y=4x \]
Step 4: Find $y$ when $x=5$.
\[ y=4\times5 \]
\[ \boxed{y=20} \]
Remember: for direct proportion, divide $y$ by $x$ to find $k$.
\[ \boxed{k=\frac{y}{x}} \]
Inverse proportion
As one quantity increases, the other decreases.
For example, imagine 12 cakes need decorating.
- 1 person takes 12 hours.
- 2 people take 6 hours.
- 3 people take 4 hours.
- 4 people take 3 hours.
Assuming everyone works at the same rate, more people means less time.
The formula for inverse proportion is:
\[ y=\frac{k}{x} \]
Equivalently, $xy=k$.
Worked example 2
$y$ is inversely proportional to $x$.
When $x=4$, $y=6$. Find $y$ when $x=8$.
Step 1: Write the formula.
\[ y=\frac{k}{x} \]
Step 2: Find $k$.
Substitute $x=4$ and $y=6$:
\[ 6=\frac{k}{4} \]
Multiply both sides by 4:
\[ k=24 \]
Step 3: Write the equation.
\[ y=\frac{24}{x} \]
Step 4: Find $y$ when $x=8$.
\[ y=\frac{24}{8} \]
\[ \boxed{y=3} \]
Remember: for inverse proportion, multiply $x$ by $y$ to find $k$.
\[ \boxed{k=xy} \]
Compare the two types
| Direct proportion | Inverse proportion |
|---|---|
| $y=kx$ | $y=\dfrac{k}{x}$ |
| $k=\dfrac{y}{x}$ | $k=xy$ |
| Both change in the same direction | They change in opposite directions |
| Double $x$, double $y$ | Double $x$, halve $y$ |
The exam tip to remember
- Direct: divide to find the constant, $k=y\div x$.
- Inverse: multiply to find the constant, $k=xy$.
- Always use the given values to find $k$ first, then substitute the new value into your formula.