Direct and Inverse Proportion
Proportion helps us understand how quantities relate to each other. From scaling recipes to calculating journey times, it’s a key skill for solving a huge range of real-world problems where things change in a predictable way. ⚖️
Direct vs. Inverse Proportion
Direct Proportion
As one quantity increases, the other increases at the same rate. If one doubles, the other doubles.
The graph is a straight line through the origin.
Inverse Proportion
As one quantity increases, the other decreases. If one doubles, the other halves.
The graph is a hyperbolic curve that never touches the axes.
The 3-Step Method to Solve Proportion Problems
1. Write the Equation
Start with the general equation for the relationship ($y = kx$ for direct, or $y = \frac{k}{x}$ for inverse).
2. Find the Constant ($k$)
Substitute the pair of values you know into the equation and solve for $k$.
3. Use the Full Equation
Write the specific equation with your value of $k$ and use it to find the final answer.
Worked Examples
Example 1: Direct Proportion 🚲
It costs £24 to hire a bike for 3 hours. How much does it cost for 7 hours?
- Equation: Cost ($C$) is directly proportional to hours ($H$), so $C = kH$.
- Find $k$: $24 = k \times 3 \implies k = 8$.
- Full equation ($C = 8H$): For 7 hours, $C = 8 \times 7 = 56$.
Answer: £56
Example 2: Inverse Proportion 👷
It takes 4 people 30 minutes to dig a trench. How long would it take 6 people?
- Equation: Time ($T$) is inversely proportional to people ($P$), so $T = \frac{k}{P}$.
- Find $k$: $30 = \frac{k}{4} \implies k = 120$.
- Full equation ($T = \frac{120}{P}$): For 6 people, $T = \frac{120}{6} = 20$.
Answer: 20 minutes
Practice Questions
- $y$ is directly proportional to $x$. When $x=4$, $y=28$. Find the value of $y$ when $x=7$.
- $A$ is inversely proportional to $B$. When $A=5$, $B=6$. Find the value of $A$ when $B=15$.
- The time taken ($T$ minutes) to fill a pool is inversely proportional to the rate of flow ($R$ litres/min). It takes 60 minutes at 20 litres/min. How long would it take at 30 litres/min?
Show Answers
- Working: $y = kx \implies 28 = 4k \implies k = 7$. Equation: $y = 7x$. When $x = 7$: $y = 49$.
Answer: $y = 49$. - Working: $A = k/B \implies 5 = k/6 \implies k = 30$. Equation: $A = 30/B$. When $B = 15$: $A = 2$.
Answer: $A = 2$. - Working: $T = k/R \implies 60 = k/20 \implies k = 1200$. Equation: $T = 1200/R$. When $R = 30$: $T = 40$.
Answer: 40 minutes.
Need Help with Proportion?
A qualified GCSE maths tutor can guide you through direct and inverse proportion, ratio, and every other algebra topic on the syllabus.
Find a GCSE Maths Tutor
