Proportionality
Proportionality helps us understand how quantities relate to each other. From scaling recipes to calculating taxi fares, 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 ($y = kx$ 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
A 5-mile taxi journey costs £8. Cost ($C$) is directly proportional to distance ($d$). Find the cost of a 12-mile journey.
- Equation: $C = kd$.
- Find $k$: $8 = k \times 5 \implies k = 1.6$.
- Full equation: $C = 1.6d$.
- Solve: $C = 1.6 \times 12 = 19.2$.
Answer: £19.20
Example 2: Inverse Proportion
It takes 4 people 30 minutes to dig a trench. How long would it take 6 people?
- Equation: $T = \dfrac{k}{P}$.
- Find $k$: $30 = \dfrac{k}{4} \implies k = 120$.
- Full equation: $T = \dfrac{120}{P}$.
- Solve: $T = \dfrac{120}{6} = 20$.
Answer: 20 minutes
What You Need to Know for GCSEs
In your exams, you’ll need to be able to:
- Identify direct and inverse proportion from statements, tables, or graphs.
- Formulate equations using the constant of proportionality ($k$).
- Calculate the constant ($k$) by substituting in given values.
- Use the derived equation to find unknown values.
- Solve real-life word problems.
- Clearly show your working to secure method marks.
Practice Questions
- $y$ is directly proportional to $x$. When $x=4$, $y=28$. Find $y$ when $x=7$.
- $A$ is inversely proportional to $B$. When $A=5$, $B=6$. Find $A$ when $B=15$.
- The time ($T$ 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 at 30 litres/min?
Show Answers
- Working: $y = kx \implies 28 = 4k \implies k = 7$. When $x = 7$: $y = 7 \times 7 = 49$.
Answer: 49. - Working: $A = k/B \implies 5 = k/6 \implies k = 30$. When $B = 15$: $A = 30/15 = 2$.
Answer: 2. - Working: $T = k/R \implies 60 = k/20 \implies k = 1200$. When $R = 30$: $T = 1200/30 = 40$.
Answer: 40 minutes.
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