What is the difference between ratio and proportion in Edexcel IGCSE Mathematics?

In Edexcel IGCSE Mathematics, a ratio is a way to compare two or more quantities by showing the relative sizes of each quantity. For example, the ratio 3:2 indicates that for every 3 units of one quantity, there are 2 units of another. Proportion, on the other hand, refers to the equality of two ratios. It is a statement that two ratios are equivalent, such as 3:2 = 6:4, meaning the two ratios have the same relationship.

How do you solve ratio problems in the Edexcel IGCSE curriculum?

To solve ratio problems in the Edexcel IGCSE curriculum, first express the given quantities in ratio form. Simplify the ratio if possible by dividing both terms by their greatest common divisor. If the problem involves finding an unknown, set up a proportion and solve for the unknown by cross-multiplying and dividing. Always check your solution to ensure it satisfies the original problem conditions.

What are some common applications of ratio and proportion in real-life scenarios?

Ratio and proportion are widely used in real-life scenarios such as cooking, where ingredients are mixed in specific ratios; in map reading, where distances are represented proportionally; and in financial contexts, like calculating interest rates or comparing prices. Understanding these concepts helps in making informed decisions based on quantitative relationships.

How can I simplify a ratio in Edexcel IGCSE Mathematics?

To simplify a ratio in Edexcel IGCSE Mathematics, divide each term of the ratio by their greatest common divisor (GCD). For example, to simplify the ratio 8:12, find the GCD of 8 and 12, which is 4. Divide both terms by 4 to get the simplified ratio of 2:3. Simplifying ratios makes them easier to work with and compare.

What techniques can be used to solve proportion problems in the Edexcel IGCSE exam?

In the Edexcel IGCSE exam, solving proportion problems typically involves setting up an equation where two ratios are equal. Use cross-multiplication to solve for the unknown variable. Additionally, understanding unitary method, where you find the value of a single unit and then multiply to find the desired quantity, can also be helpful. Practice these techniques to become proficient in solving proportion problems.

Why is "Writing a ratio with mixed units" a common mistake in Ratio and Proportion?

Why it happens: Question gives one quantity in cm, another in m, and student doesn't convert. How to avoid it: ALWAYS convert to the same unit BEFORE writing the ratio. Show this step explicitly: 1.5 m = 150 cm. Source: 4MA1/1H — examiner reports 2022-2024

Why is "Treating an inverse proportion as direct" a common mistake in Ratio and Proportion?

Why it happens: Both seem to scale, but in opposite directions. How to avoid it: Test the relationship: does increasing one INCREASE or DECREASE the other? Direct = same direction; Inverse = opposite. Check with extreme values. Source: 4MA1/2H May/Jun 2023 — examiner report Q11

Why is "In ratio $a : b$, calling the fraction $\dfrac{a}{b}$ the 'fraction of the total'" a common mistake in Ratio and Proportion?

Why it happens: Looks intuitive — but the denominator should be a + b, not b. How to avoid it: a/a+b for the fraction of the total that's a. The total parts is the SUM, not just one side.

Why is "Trying to scale up directly without finding the constant of variation" a common mistake in Ratio and Proportion?

Why it happens: Students see 'when x = 4, y = 32' and try shortcuts. How to avoid it: Always: (1) state y = kx^n, (2) find k, (3) use the equation. The structured method prevents errors and earns method marks.

Numbers and the Number SystemEdexcel IGCSE Mathematics (4MA1)

Ratio and Proportion

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

Previous Next

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Page 1 / 0

Detailed Study Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Ratio and Proportion, ready to print or save offline.

Ratio and Proportion Study Notes — Edexcel IGCSE 4MA1 Higher Tier (2026 onwards)

Sharing amounts, simplifying ratios, direct and inverse proportion. The technique behind every recipe scaling, currency conversion, and physics calculation in 4MA1.

What you’ll learn

Mapped to the Cambridge IGCSE 4MA1 syllabus (2026 onwards).

1.8 — Use ratio notation; simplify and find equivalent ratios.
1.8 — Divide a quantity in a given ratio.
1.9 — Solve direct and inverse proportion problems.
1.9 — Find the constant of variation.
1.9 — Use proportion in real-life contexts (best buys, recipes).

Working with ratios

Ratios compare quantities. Always simplify, always check units.

Notation. A ratio is written $a : b$ (read 'a to b') or $a : b : c$ for three quantities.

Simplify by dividing each part by their HCF.

Example: $40 : 60 : 100$ . HCF $= 20$ . Simplify: $2 : 3 : 5$ .

Convert to common units before simplifying.

$40$ cm : $1.5$ m. Convert: $40 : 150$ . Simplify: $4 : 15$ .

Equivalent ratios. Multiply or divide all parts by the same number.

$2 : 3 = 4 : 6 = 6 : 9 = 100 : 150$ .

Sharing in a ratio.

To divide an amount $T$ in ratio $a : b : c$ :

Total parts: $p = a + b + c$ .
One part: $T / p$ .
Each share: multiply 'one part' by the relevant ratio number.

Example: Share £ $240$ in $2 : 3 : 7$ .

Parts: $12$ .
One part: $20$ .
Shares: £ $40, £60, £140$ .

Ratio to fraction.

In a ratio $a : b$ , the fraction that is $a$ is $\dfrac{a}{a+b}$ . NOT $\dfrac{a}{b}$ .

Example: Ratio of girls to boys $5 : 7$ . Fraction of girls = $\dfrac{5}{5+7} = \dfrac{5}{12}$ .

Worked qualitative. A recipe makes 12 cookies using 240g flour. How much flour for 30 cookies?

Use direct proportion: flour scales with cookies.
Multiplier: $30 / 12 = 2.5$ .
Flour: $240 \times 2.5 = 600$ g.

Edexcel tip. When dividing in a ratio, ALWAYS check your shares add to the total. Mark schemes credit method marks for setting up correctly.

$a : b : c$ — same units throughout.
Simplify by HCF.
Sharing: parts $\to$ one part $\to$ each share.
Ratio to fraction: denominator is sum.

Direct proportion

$y \propto x^{n}$ means $y = kx^{n}$ . Find $k$ first, then apply.

Definition. $y$ is directly proportional to $x^{n}$ (often $n = 1$ ) if:

$y = kx^{n}$

for some constant $k$ (called the constant of variation).

The standard method.

Set up: $y = kx^{n}$ (with the right power).
Find $k$ : substitute the given values.
Use: substitute back to answer the question.

Worked example. $y$ is directly proportional to $x^{2}$ . When $x = 4$ , $y = 32$ . Find $y$ when $x = 6$ .

$y = kx^{2}$ .
$32 = k \times 16 \implies k = 2$ .
$y = 2x^{2}$ . When $x = 6$ : $y = 2 \times 36 = 72$ .

Check by ratio. $x$ goes from $4$ to $6$ (multiplier $1.5$ ). Since $y \propto x^{2}$ , $y$ multiplies by $1.5^{2} = 2.25$ . Check: $32 \times 2.25 = 72$ . ✓

Common forms:

Statement	Equation
$y$ proportional to $x$	$y = kx$
$y$ proportional to $x^{2}$	$y = kx^{2}$
$y$ proportional to $\sqrt{x}$	$y = k\sqrt{x}$
$y$ proportional to $x^{3}$	$y = kx^{3}$

Worked qualitative. Why does the surface area of a sphere scale as $r^{2}$ but its volume as $r^{3}$ ?

Surface area is $4\pi r^{2}$ — area scales as $r^{2}$ .
Volume is $\frac{4}{3}\pi r^{3}$ — volume scales as $r^{3}$ .
Doubling radius: SA up $4\times$ , volume up $8\times$ .
This is why bigger animals have proportionally smaller surface area to volume — relevant in biology.

Edexcel tip. State the proportion equation explicitly. 'Direct proportion' as a phrase isn't enough — write $y = kx$ or $y = kx^{2}$ .

$y \propto x^{n}$ → $y = kx^{n}$ .
Find $k$ from given data.
Substitute back to find unknown.
Common: $n = 1, 2, 1/2$ .

Inverse proportion

$y \propto 1/x^{n}$ means $y = k/x^{n}$ . Same method as direct.

Definition. $y$ is inversely proportional to $x^{n}$ if:

$y = \dfrac{k}{x^{n}}$

The product $y \times x^{n}$ is constant (equal to $k$ ).

Doubling $x$ halves $y$ (if $n = 1$ ). Doubling $x$ quarters $y$ (if $n = 2$ ).

Worked example. $P$ is inversely proportional to $Q^{2}$ . When $Q = 3$ , $P = 8$ . Find $P$ when $Q = 6$ .

$P = k / Q^{2}$ .
$8 = k / 9 \implies k = 72$ .
$P = 72 / Q^{2}$ . When $Q = 6$ : $P = 72 / 36 = 2$ .

Common contexts:

Scenario	Relationship
Time to complete job vs workers	Inverse
Speed vs time (fixed distance)	Inverse
Volume vs pressure (gas at constant temp)	Inverse
Force of gravity vs distance²	Inverse

Worked qualitative. A car travels a fixed distance. If the speed doubles, what happens to the time?

Distance = speed × time.
Fixed distance: time $=$ distance/speed.
Time inversely proportional to speed.
Doubling speed halves time.

Edexcel tip. Once you find $k$ , your equation is fully determined. Don't recompute — just plug in.

$y = k/x^{n}$ .
Same method: state, find $k$ , use.
Doubling $x$ halves (or quarters) $y$ .
Time vs workers, speed vs time.

Real-life proportion — best buys, exchange rates

Use rate per unit. Compare like-for-like.

Best-buy problems. Find unit price (£ per kg, p per litre) for each option. The lowest is best.

Example: Brand A: $500$ g for £ $3.00$ . Brand B: $750$ g for £ $4.20$ .

A: $300$ p / $500$ g $= 0.60$ p per g.
B: $420$ p / $750$ g $= 0.56$ p per g.
B is cheaper per gram.

Currency conversion.

Use the exchange rate as a multiplier.

Example: $1$ GBP $= 1.25$ USD. Convert £ $80$ to USD.

$80 \times 1.25 =$ 100$ USD.

Reverse: convert $\$ 100$ to GBP.

$100 / 1.25 = £80$ .

Recipes / scaling.

Use direct proportion. Multiply all ingredients by the same factor.

Example: Recipe for $4$ people uses $200$ g pasta. For $7$ people:

Multiplier: $7/4 = 1.75$ .
Pasta needed: $200 \times 1.75 = 350$ g.

Worked qualitative. Why are 'unit prices' the fairest comparison for best buys?

They normalise to the same quantity.
Removes confusion of different package sizes.
Universally comparable.
Many shops display unit price by law for this reason.

Edexcel tip. For best buys, ALWAYS convert to the same unit. Cents per gram, pence per litre, etc. Don't compare different units (£/kg vs p/g without conversion).

Best buy = lowest unit price.
Convert to same units.
Currency: rate as multiplier.
Recipes: multiply all by scaling factor.

How it’s examined

Ratio and proportion appear on EVERY 4MA1 paper, often for 4-6 marks. Typical: share an amount (3 marks), direct or inverse proportion problem (3-5 marks), best-buy comparison (3 marks). Examiner reports flag (1) treating inverse as direct, (2) skipping the 'find $k$ ' step, and (3) ratio with mixed units.

Sources: Pearson Edexcel International GCSE Mathematics A 4MA1 specification (2026 onwards); 4MA1 Examiner Reports — Jan and May/Jun 2022-2024 series; Past Papers — 4MA1/1H Jan 2024, 4MA1/2H May/Jun 2023. Last reviewed 2026-05-07.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Ratio and Proportion, ready to print or save as PDF.

Step-by-step worked examples — Ratio and Proportion

Step-by-step solutions to past-paper-style questions on ratio and proportion, written exactly the way a tutor would explain them at the board.

1Share an amount in a ratio
Foundation• sharing, ratio
Question
Divide £ $420$ in the ratio $2 : 3 : 5$ .
Step-by-step solution
1. Step 1
  Add the parts: $2 + 3 + 5 = 10$ parts total.
2. Step 2
  One part: $\dfrac{420}{10} = £42$ .
3. Step 3
  Multiply by each part: $2 \times 42 = £84$ , $3 \times 42 = £126$ , $5 \times 42 = £210$ .
4. Step 4
  Check: $84 + 126 + 210 = £420$ . ✓
Answer
£ $84$ : £ $126$ : £ $210$
Examiner tip
ALWAYS check your answer adds back to the total. Mark schemes award M1 for finding 'one part' and A1 for the three correct shares.
2Direct proportion problem
Foundation• direct proportion
Question
$5$ workers can build a wall in $6$ days. Working at the same rate, how long would $3$ workers take?
Step-by-step solution
1. Step 1
  Use total work-days: $5$ workers $\times 6$ days $= 30$ worker-days.
2. Step 2
  $30$ worker-days $\div 3$ workers $= 10$ days.
3. Step 3
  More workers → less time (INVERSE proportion). Fewer workers → more time. Result $10 > 6$ days makes sense.
Answer
$10$ days
Examiner tip
Check whether the relationship is DIRECT (more = more) or INVERSE (more = less). Mistaking the type loses all marks.
3Direct proportion using $y \propto x$
Higher• Adapted from 4MA1/1H Jan 2024 Q15• direct proportion, constant of variation
Question
$y$ is directly proportional to $x^{2}$ . When $x = 4$ , $y = 32$ . Find $y$ when $x = 6$ .
Step-by-step solution
1. Step 1
  $y \propto x^{2}$ means $y = kx^{2}$ for some constant $k$ .
2. Step 2
  Substitute the known values to find $k$ : $32 = k \times 4^{2} = 16k$ , so $k = 2$ .
3. Step 3
  The relationship is $y = 2x^{2}$ .
4. Step 4
  When $x = 6$ : $y = 2 \times 6^{2} = 2 \times 36 = 72$ .
Answer
$y = 72$
Examiner tip
Always state $y = kx^{n}$ , find $k$ , then use the equation. Edexcel mark schemes reward this structured approach.
4Inverse proportion
Higher• Adapted from 4MA1/2H May/Jun 2023 Q11• inverse proportion
Question
$P$ is inversely proportional to $Q^{2}$ . When $Q = 3$ , $P = 8$ . Find $P$ when $Q = 6$ .
Step-by-step solution
1. Step 1
  $P \propto \dfrac{1}{Q^{2}}$ means $P = \dfrac{k}{Q^{2}}$ .
2. Step 2
  Find $k$ : $8 = \dfrac{k}{9}$ , so $k = 72$ .
3. Step 3
  Equation: $P = \dfrac{72}{Q^{2}}$ .
4. Step 4
  When $Q = 6$ : $P = \dfrac{72}{36} = 2$ .
Answer
$P = 2$
Examiner tip
Inverse proportion: $P = k/Q^{n}$ . Doubling $Q$ doesn't halve $P$ when there's a power involved. ALWAYS find $k$ first.
5Ratio to fraction conversion
Foundation• ratio, fraction
Question
In a school, the ratio of girls to boys is $5 : 7$ . What FRACTION of the school are girls?
Step-by-step solution
1. Step 1
  Total parts: $5 + 7 = 12$ .
2. Step 2
  Girls are $5$ of $12$ parts: fraction = $\dfrac{5}{12}$ .
Answer
$\dfrac{5}{12}$
Examiner tip
Add ALL parts to get the denominator. Common error: using the OTHER side of the ratio (writing $\frac{5}{7}$ instead of $\frac{5}{12}$ ).
6Simplify a ratio with units
Foundation• simplifying ratio
Question
Express the ratio $40 \text{ cm} : 1.5 \text{ m}$ in its simplest form.
Step-by-step solution
1. Step 1
  Convert to the same units: $1.5$ m $= 150$ cm.
2. Step 2
  Ratio: $40 : 150$ . Simplify by dividing both by HCF $= 10$ .
3. Step 3
  $40 \div 10 : 150 \div 10 = 4 : 15$ .
Answer
$4 : 15$
Examiner tip
Convert to a COMMON UNIT before simplifying. Skipping this step is the #1 error — write 'in cm' as the first line of working.

Key Formulae — Ratio and Proportion

The formulae you need to memorise for ratio and proportion on the Pearson Edexcel IGCSE 4MA1 paper, with every variable defined in plain English and a note on when to use it.

Direct proportion
$y \propto x^{n} \iff y = kx^{n}$
$k$
constant of variation (proportionality constant)
$n$
power of $x$ (often $1$ , sometimes $2$ or higher)
When to use
When two variables increase together. Find $k$ from given data, then use the equation.
Example
$y \propto x^{2}$ , $y = 32$ when $x = 4$ . Then $k = 2$ and $y = 2x^{2}$ .
Inverse proportion
$y \propto \dfrac{1}{x^{n}} \iff y = \dfrac{k}{x^{n}}$
$k$
constant of variation
When to use
When one increases as the other decreases. Find $k$ , then use the equation.
Example
$P \propto \dfrac{1}{Q^{2}}$ , $P = 8$ when $Q = 3$ . Then $k = 72$ and $P = 72 / Q^{2}$ .
Sharing in a ratio
$\text{Each share} = \dfrac{\text{share's parts}}{\text{total parts}} \times \text{total amount}$
$\text{total parts}$
sum of all parts in the ratio
When to use
Whenever an amount is to be divided in a given ratio. Use 'one part' = total/(sum of parts).
Example
Share £ $420$ in $2 : 3 : 5$ ( $10$ parts): each part $= £42$ , so shares are $£84, £126, £210$ .

Key Definitions and Keywords — Ratio and Proportion

Definitions to memorise and the exact keywords mark schemes credit for ratio and proportion answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IGCSE 4MA1 sitting.

Ratio
Examiner keyword
A way of comparing two or more quantities of the same kind, written $a : b$ or $a : b : c$ .
Example
Recipe ratio of flour to sugar is $3 : 1$ .
Proportion
Examiner keyword
A statement that two ratios (or rates) are equal: $a : b = c : d$ .
Example
If $2 : 5 = 8 : x$ , solving gives $x = 20$ .
Direct proportion
Examiner keyword
$y$ is directly proportional to $x$ (written $y \propto x$ ) if $y = kx$ for some constant $k$ . Doubling $x$ doubles $y$ .
Example
Cost of fruit at £ $2$ /kg: cost $\propto$ weight.
Inverse proportion
Examiner keyword
$y$ is inversely proportional to $x$ (written $y \propto \dfrac{1}{x}$ ) if $y = \dfrac{k}{x}$ for some constant $k$ . Doubling $x$ halves $y$ .
Example
Time to complete a job is inversely proportional to number of workers.
Constant of variation (or constant of proportionality)
The fixed value $k$ in a proportion equation $y = kx^{n}$ or $y = k/x^{n}$ . Found from given data.

Common Mistakes and Misconceptions — Ratio and Proportion

The traps other students keep falling into on ratio and proportion questions — taken from recent Pearson Edexcel IGCSE 4MA1 examiner reports and mark schemes — and how to avoid them.

✕Writing a ratio with mixed units
4MA1/1H — examiner reports 2022-2024
Why it happens
Question gives one quantity in cm, another in m, and student doesn't convert.
How to avoid it
ALWAYS convert to the same unit BEFORE writing the ratio. Show this step explicitly: $1.5$ m $= 150$ cm.
✕Treating an inverse proportion as direct
4MA1/2H May/Jun 2023 — examiner report Q11
Why it happens
Both seem to scale, but in opposite directions.
How to avoid it
Test the relationship: does increasing one INCREASE or DECREASE the other? Direct = same direction; Inverse = opposite. Check with extreme values.
✕In ratio $a : b$ , calling the fraction $\dfrac{a}{b}$ the 'fraction of the total'
Why it happens
Looks intuitive — but the denominator should be $a + b$ , not $b$ .
How to avoid it
$\dfrac{a}{a+b}$ for the fraction of the total that's $a$ . The total parts is the SUM, not just one side.
✕Trying to scale up directly without finding the constant of variation
Why it happens
Students see 'when $x = 4$ , $y = 32$ ' and try shortcuts.
How to avoid it
Always: (1) state $y = kx^{n}$ , (2) find $k$ , (3) use the equation. The structured method prevents errors and earns method marks.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Ratio and Proportion — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Ratio and Proportion

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Share an amount in a ratio

2Direct proportion problem

3Direct proportion using y∝xy \propto xy∝x

4Inverse proportion

5Ratio to fraction conversion

6Simplify a ratio with units

Direct proportion

Inverse proportion

Sharing in a ratio

Ratio

Proportion

Direct proportion

Inverse proportion

Constant of variation (or constant of proportionality)

✕Writing a ratio with mixed units

✕Treating an inverse proportion as direct

✕In ratio a:ba : ba:b, calling the fraction ab\dfrac{a}{b}ba​ the 'fraction of the total'

✕Trying to scale up directly without finding the constant of variation

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Ratio and Proportion — frequently asked questions

3Direct proportion using $y \propto x$

✕In ratio $a : b$ , calling the fraction $\dfrac{a}{b}$ the 'fraction of the total'