# Proportion (Edexcel IGCSE Mathematics A Higher Tier)

> Edexcel International GCSE Mathematics A · 4MA1 2016
> Source: https://www.owlsprep.com/study/edexcel-igcse-math-a-s2-proportion/

This guide covers all higher-tier proportion content for Edexcel IGCSE Maths A (4MA1), including setting up equations, solving for constant $k$, and linking relationships to graph shapes.

**Prerequisites:** [Rearranging formulae](https://www.owlsprep.com/study/edexcel-igcse-math-a-s2-rearranging-formulae/); [Plotting linear and non-linear graphs](https://www.owlsprep.com/study/edexcel-igcse-math-a-s3-graphs-basics/)

## Learning objectives

- Distinguish between all 8 allowed direct and inverse proportion relationships
- Set up proportion equations using the constant of proportionality $k$
- Calculate $k$ from given data and solve for unknown values
- Match proportion equations to their corresponding graph shapes
- Solve full-mark exam-style proportion problems accurately

## Direct Proportion Relationships

**Direct Proportion** — Two positive quantities are directly proportional if as one increases, the other increases by a constant factor. This is written as an equation: $y = kf(x)$, where $k$ is the non-zero constant of proportionality.

*Notation:* $y \propto f(x)$

*Example:* If $y$ is directly proportional to the square of $x$, then $y = kx^2$

- Allowed direct proportion relationships: $y \propto x$, $y \propto x^2$, $y \propto x^3$, $y \propto \sqrt{x}$
- All direct proportion graphs pass through the origin (0,0)
- Only $y \propto x$ gives a straight line graph; other direct relationships give upward-curving lines through the origin

**Worked example:** The cost $C$ of producing metal discs is directly proportional to the square of the radius $r$ of the disc. When $r = 4$ cm, $C = \pounds{2.40}$. Find the cost of a disc with radius 7 cm.

1. Write the proportion statement: $C \propto r^2$
2. Introduce constant $k$ to form an equation:

   $$C = kr^2$$
3. Substitute given values to solve for $k$: $2.40 = k(4)^2 = 16k$, so $k = 2.40 / 16 = 0.15$
4. Write the completed equation: $C = 0.15r^2$
5. Substitute $r=7$: $C = 0.15(49) = 7.35$
6. Final answer: $\pounds{7.35}$

> **Exam tip:** Underline the power of x in the question (e.g., square, cube, square root) to avoid writing the wrong function in your proportion equation.

*Calculator:* allowed

## Inverse Proportion Relationships

**Inverse Proportion** — Two positive quantities are inversely proportional if as one increases, the other decreases by a constant factor. This is written as an equation: $y = \frac{k}{f(x)}$, where $k$ is the non-zero constant of proportionality.

*Notation:* $y \propto \frac{1}{f(x)}$

*Example:* If $y$ is inversely proportional to the cube of $x$, then $y = \frac{k}{x^3}$

- Allowed inverse proportion relationships: $y \propto 1/x$, $y \propto 1/x^2$, $y \propto 1/x^3$, $y \propto 1/\sqrt{x}$
- All inverse proportion graphs are decreasing curves in the first quadrant
- Inverse graphs never touch the x or y axes, as dividing by zero is undefined

**Worked example:** The time $t$ taken to fill a swimming pool is inversely proportional to the square root of the number of pumps $p$ used. When 16 pumps are used, the pool fills in 90 minutes. How many pumps are needed to fill the pool in 60 minutes?

1. Write the proportion statement: $t \propto \frac{1}{\sqrt{p}}$
2. Introduce constant $k$ to form an equation:

   $$t = \frac{k}{\sqrt{p}}$$
3. Substitute given values to solve for $k$: $90 = \frac{k}{\sqrt{16}} = \frac{k}{4}$, so $k = 360$
4. Write the completed equation: $t = \frac{360}{\sqrt{p}}$
5. Substitute $t=60$: $60 = \frac{360}{\sqrt{p}}$ → $\sqrt{p} = 6$ → $p = 36$
6. Final answer: 36 pumps

> **Exam tip:** Always double-check that you have included the reciprocal for inverse proportion statements, as this is the most common error on this topic.

*Calculator:* allowed

## Linking Proportion Equations to Graphs

> **info**
>
> You will often be asked to match proportion equations to graph shapes in 1-2 mark multiple choice or short answer questions. Use the origin/intercept rule first to eliminate wrong answers quickly.

**Worked example:** State the key features of the graph of $y$ inversely proportional to the square of $x$, for positive values of $x$ and $y$.

1. First write the full equation: $y = \frac{k}{x^2}$, where $k$ is positive.
2. Check for intercepts: When $x=0$, $y$ is undefined, so no y-intercept. When $y=0$, $1/x^2 = 0$ which has no solution, so no x-intercept.
3. Check shape: As $x$ increases, $y$ decreases, so the graph is a downward-sloping curve in the first quadrant.
4. Final key features: No intercepts, decreasing curve approaching but never touching the x and y axes.

**Check your understanding**

1. Which of the following graphs represents $y \propto x^3$?

   - Straight line through origin
   - Upward curving line through origin
   - Decreasing curve with no intercepts
   - Horizontal line

   *Answer:* Upward curving line through origin

   *Why:* Correct: All direct power relationships other than linear give upward curving lines through the origin.

> **Exam tip:** If asked to sketch a proportion graph, always label the axes and add arrows at the ends of the curve to show it continues indefinitely.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to introduce the constant of proportionality $k$ and jumping straight to calculating values.
  - Why it fails: Examiners award 1 mark for explicitly stating the proportion equation with $k$, so you lose marks even if your final answer is correct.
  - Correct: Always write $y = kf(x)$ (direct) or $y = k/f(x)$ (inverse) immediately after your initial proportion statement.
- **Wrong:** Mixing up direct and inverse proportion, e.g., writing $y = kx$ for "y is inversely proportional to x".
  - Why it fails: This leads to completely incorrect values for $k$ and all subsequent answers.
  - Correct: Underline the phrase "directly proportional" or "inversely proportional" in the question, and add the reciprocal for inverse statements immediately.
- **Wrong:** Using rounded decimal values for $k$ instead of exact fractions or integers.
  - Why it fails: Rounding $k$ early leads to accumulated errors in final answers, which may fall outside the allowed tolerance range.
  - Correct: Keep $k$ exact through all calculations, only rounding the final answer if explicitly asked.
- **Wrong:** Assuming all direct proportion graphs are straight lines.
  - Why it fails: Only $y \propto x$ gives a straight line; $y \propto x^2, x^3, \sqrt{x}$ give curved lines through the origin.
  - Correct: Check the power of $x$ in the relationship to determine if the graph is linear or curved.
- **Wrong:** Drawing inverse proportion graphs that touch the x or y axis.
  - Why it fails: For inverse relationships, $x$ or $y$ can never be zero (division by zero is undefined), so intercepts are impossible.
  - Correct: Draw inverse proportion curves approaching but never touching the axes, with arrows at the ends.

## Cheatsheet

| Relationship Type | Proportion Statement | Equation with k | Key Graph Features |
| --- | --- | --- | --- |
| Direct Linear | $y \propto x$ | $y = kx$ | Straight line through origin, gradient k |
| Direct Square | $y \propto x^2$ | $y = kx^2$ | Upward curving line through origin |
| Direct Cube | $y \propto x^3$ | $y = kx^3$ | Steeper upward curving line through origin |
| Direct Square Root | $y \propto \sqrt{x}$ | $y = k\sqrt{x}$ | Flattening upward curve through origin |
| Inverse Linear | $y \propto 1/x$ | $y = k/x$ | Decreasing curve, no intercepts |
| Inverse Square | $y \propto 1/x^2$ | $y = k/x^2$ | Steeper decreasing curve, no intercepts |
| Inverse Cube | $y \propto 1/x^3$ | $y = k/x^3$ | Very steep decreasing curve, no intercepts |
| Inverse Square Root | $y \propto 1/\sqrt{x}$ | $y = k/\sqrt{x}$ | Flattening decreasing curve, no intercepts |

## What's next

Now that you have mastered proportion for Edexcel IGCSE Maths A Higher Tier, you can apply these skills to solve real-world context problems involving rates, area, volume, and scaling that often appear alongside proportion questions. Proportion questions typically make up 5-8% of the higher tier paper, so consistent practice of past paper questions will help you avoid common errors and secure full marks on this topic. You should also practice combining proportion with rearranging formulae and graph interpretation to tackle the most challenging 4-5 mark exam questions.

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