Ratio and Proportion · Study Notes · Edexcel IGCSE 4MA1 Higher

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

At a glance

Sharing: total parts $=$ sum of ratio. Each part $=$ total $\div$ parts.
Simplify: divide all parts by HCF. Convert to common units first.
Direct: $y \propto x^{n}$ → $y = kx^{n}$ .
Inverse: $y \propto 1/x^{n}$ → $y = k/x^{n}$ .
Method: state $y = kx^{n}$ , find $k$ , then use the equation.
Ratio to fraction: $a : b$ → fraction of $a$ is $\dfrac{a}{a+b}$ .

What you’ll learn

Mapped to the Pearson Edexcel 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

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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: £ $40, £60, £140$ .

Add the ratio numbers to get total parts, find one part, then scale up each share.

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

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$y \propto x^{n}$ 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: $32 \times 2.25 = 72$ . ✓

A directly proportional relationship is a straight line through the origin.

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

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$y \propto 1/x^{n}$ 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 $n = 2$ ).

As x increases y decreases; the curve approaches but never reaches the axes.

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

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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.

Quick recap

Ratio: same units, simplify by HCF.
Sharing: total parts → one part → shares.
Direct: $y = kx^{n}$ .
Inverse: $y = k/x^{n}$ .
Method: state, find $k$ , use.
Best buy: lowest unit price.

Memorise this

Verbatim phrases and definitions Edexcel mark schemes credit.

Direct proportion: $y = kx^{n}$ .
Inverse proportion: $y = k/x^{n}$ .
Sharing: each share $=$ (its parts $/$ total parts) $\times$ amount.
Ratio to fraction: denominator is the SUM of parts.
Standard method: state, find $k$ , use.

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.

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
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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
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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
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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
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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
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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
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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.
7From a fraction of a group to a ratio [1 mark]
Getting started• Paper 1H/2H style — 1 mark• ratio, fractions
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Question
In a class, $\frac{3}{7}$ of the students are boys. The rest are girls.

Write down the ratio of boys to girls. [1]
Step-by-step solution
1. Step 1
  B1 If $\frac{3}{7}$ are boys, then $\frac{4}{7}$ are girls. Out of every $7$ students, $3$ are boys and $4$ are girls.
  $\text{boys} : \text{girls} = 3 : 4$
Answer
$3 : 4$
Examiner tip
Examiner tip: The denominator of the fraction is the TOTAL number of parts; the ratio splits that total into its two groups.

Common mistake: Writing $3 : 7$ — that compares boys with the whole class, not boys with girls.
8Direct proportion — cost of different masses [2 marks]
Getting started• Paper 1H/2H style — 2 marks• direct proportion, unitary method
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Question
The cost of cheese is directly proportional to its mass.

$350$ g of cheese costs £4.55

Work out the cost of

(a) $200$ g of cheese [1]

(b) $1.2$ kg of cheese [1]
Step-by-step solution
1. Step 1
  Find the cost of $1$ gram, in pence.
  $455 \div 350 = 1.3 \text{ p per g}$
2. Step 2
  B1 (a) Multiply by $200$ .
  $200 \times 1.3 = 260 \text{ p} = 2.60 \text{ pounds}$
3. Step 3
  B1 (b) Convert to grams: $1.2$ kg $= 1200$ g.
  $1200 \times 1.3 = 1560 \text{ p} = 15.60 \text{ pounds}$
Answer
(a) £2.60 (b) £15.60
Examiner tip
Examiner tip: Finding the cost of $1$ g (the unitary method) gives one multiplier that works for every mass asked about.

Common mistake: Using $1.2$ instead of $1200$ in (b) and getting about $2$ p. Always convert kilograms to grams when the rate is per gram.
9Sharing when one share is a fraction of the other [3 marks]
Building confidence• Paper 1H/2H style — 3 marks• sharing, ratio, fractions
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Question
Ed and Fay share £240

Ed receives $\frac{2}{3}$ of the amount that Fay receives.

Work out how much each of them receives. [3]
Step-by-step solution
1. Step 1
  M1 For every £3 Fay gets, Ed gets £2, so the ratio Ed : Fay is $2 : 3$
  $\text{Ed} : \text{Fay} = 2 : 3$
2. Step 2
  M1 There are $2 + 3 = 5$ parts, so find one part.
  $240 \div 5 = 48$
3. Step 3
  A1 Multiply by each number of parts.
  $\text{Ed} = 2 \times 48 = 96, \qquad \text{Fay} = 3 \times 48 = 144$
Answer
Ed £96, Fay £144
Examiner tip
Examiner tip: Check the condition, not just the total: $\frac{2}{3} \times 144 = 96$ ✓ and $96 + 144 = 240$ ✓.

Common mistake: Giving Ed $\frac{2}{3}$ of £240 (£160). The fraction is of Fay's share, not of the total.
10Best buy across two currencies [5 marks]
Stretch• Paper 1H/2H style — 5 marks• best buy, exchange rates
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Question
In the UK, a $340$ g jar of coffee costs £5.10

In Spain, a $250$ g jar of the same coffee costs €4.20

The exchange rate is £1 = €1.15

(a) Convert €4.20 into pounds. Give your answer to the nearest penny. [2]

(b) In which country is the coffee better value? You must show your working. [3]
Step-by-step solution
1. Step 1
  M1 (a) £1 buys €1.15, so divide the euros by $1.15$
  $4.20 \div 1.15 = 3.652\ldots$
2. Step 2
  A1 Round to the nearest penny: £3.65
  $3.652\ldots \approx 3.65$
3. Step 3
  M1 (b) Compare the cost of the same mass, eg $100$ g, in pounds. UK:
  $5.10 \div 3.4 = 1.50 \text{ pounds per 100 g}$
4. Step 4
  M1 Spain, using the unrounded value from (a):
  $3.652\ldots \div 2.5 = 1.460\ldots \text{ pounds per 100 g}$
5. Step 5
  A1 Spain costs less for the same mass, so it is the better value.
Answer
(a) £3.65 (b) Spain (about £1.46 per $100$ g compared with £1.50 per $100$ g in the UK)
Examiner tip
Examiner tip: Get both prices into the SAME currency and the SAME mass before comparing. Keep the unrounded value from (a) on your calculator for (b).

Common mistake: Multiplying by $1.15$ in (a) (giving £4.83) — euros to pounds divides by the rate. A second common error is comparing £5.10 with €4.20 directly, ignoring both the currency and the different jar sizes.

Question 1
Paper 1H/2H style — 1 mark1 mark
Write $0.6 : 1.5$ as a ratio of whole numbers in its simplest form. [1]
Model answer
$0.6 : 1.5 = 6 : 15$

$2 : 5$
Why this scores
Mark scheme:
- B1 for $2 : 5$ cao
Why this scores full marks: Multiplying both parts by $10$ clears the decimals, then dividing by the HCF $3$ gives the simplest form. $6 : 15$ is whole numbers but not simplest form, so it would score 0.
Question 2
Paper 1H/2H style — 1 mark1 mark
A map has a scale of $1 : 20\,000$

Write down the real distance, in metres, represented by $1$ cm on the map. [1]
Model answer
$1$ cm represents $20\,000$ cm

$20\,000 \div 100 = 200$

$200$ m
Why this scores
Mark scheme:
- B1 for $200$ cao
Why this scores full marks: The scale has no units, so $1$ cm on the map is $20\,000$ cm on the ground. Converting centimetres to metres by dividing by $100$ gives the answer in the unit asked for.
Question 3
Paper 1H/2H style — 2 marks2 marks
Orange squash is mixed with water in the ratio $1 : 6$ to make a drink.

Work out how many millilitres of squash are needed to make $1.4$ litres of the drink. [2]
Model answer
$1.4$ litres $= 1400$ ml

$1 + 6 = 7 \text{ parts}$

$1400 \div 7 = 200$

$200$ ml
Why this scores
Mark scheme:
- M1 for $1400 \div 7$ oe (or $1.4 \div 7$ )
- A1 for $200$ cao
Why this scores full marks: The $1.4$ litres is the whole drink, so it is shared into $7$ parts, not $6$ . The candidate converts to millilitres first, so the answer is already in the unit asked for.
Question 4
Paper 1H/2H style — 2 marks2 marks
Write the ratio $\frac{3}{4} : 2\frac{1}{2}$ in the form $1 : n$ [2]
Model answer
$\frac{3}{4} : \frac{5}{2}$

Multiply both parts by $4$

$3 : 10$

Divide both parts by $3$

$1 : \frac{10}{3}$

$1 : 3\frac{1}{3}$
Why this scores
Mark scheme:
- M1 for $3 : 10$ oe, or $2\frac{1}{2} \div \frac{3}{4}$
- A1 for $1 : 3\frac{1}{3}$ oe (accept $1 : \frac{10}{3}$ or $1 : 3.33$ or better)
Why this scores full marks: Writing the mixed number as an improper fraction and clearing the fractions first makes the division simple. The candidate then divides by the FIRST number, which is what the form $1 : n$ requires.
Question 5
Paper 1H/2H style — 3 marks3 marks
The length and the width of a rectangle are in the ratio $5 : 2$

The perimeter of the rectangle is $84$ cm.

Work out the area of the rectangle. [3]
Model answer
Let the length be $5k$ and the width $2k$

$2(5k + 2k) = 14k = 84 \quad\Rightarrow\quad k = 6$

Length $= 30$ cm, width $= 12$ cm

$30 \times 12 = 360$

$360 \text{ cm}^2$
Why this scores
Mark scheme:
- M1 for $84 \div 14$ ( $= 6$ ) oe
- M1 for $30 \times 12$ or $5 \times 6 \times 2 \times 6$
- A1 for $360$ cao
Why this scores full marks: The perimeter uses each side twice, so it is $14$ parts, not $7$ . Weaker scripts share $84$ in the ratio $5 : 2$ to get $60$ and $24$ , then multiply to get $1440$ .
Question 6
Paper 1H/2H style — 3 marks3 marks
Ella changes £600 into euros. The exchange rate is £1 = €1.18

On holiday she spends €450

When she comes home she changes the euros she has left back into pounds. The exchange rate is now £1 = €1.24

Work out how many pounds Ella gets. Give your answer to the nearest penny. [3]
Model answer
$600 \times 1.18 = 708 \text{ euros}$

$708 - 450 = 258 \text{ euros left}$

$258 \div 1.24 = 208.064\ldots$

£208.06
Why this scores
Mark scheme:
- M1 for $600 \times 1.18$ ( $= 708$ )
- M1 for $(708 - 450) \div 1.24$ oe
- A1 for $208.06$ cao
Why this scores full marks: Each conversion goes the right way — pounds to euros multiplies, euros to pounds divides — and the second rate is used for the second conversion. The answer is rounded to the nearest penny only at the end.
Question 7
Paper 1H/2H style — 4 marks4 marks
A recipe for $4$ portions of pasta uses $300$ g of pasta and $250$ ml of cream.

Pasta is sold in $500$ g bags costing £1.35 each.

Cream is sold in $300$ ml pots costing £1.10 each.

Ana wants to make $10$ portions. She can only buy whole bags and whole pots.

Work out the least amount Ana must spend. [4]
Model answer
Scale factor $10 \div 4 = 2.5$

$\text{pasta} = 300 \times 2.5 = 750 \text{ g} \quad\Rightarrow\quad 2 \text{ bags}$

$\text{cream} = 250 \times 2.5 = 625 \text{ ml} \quad\Rightarrow\quad 3 \text{ pots}$

$2 \times 1.35 + 3 \times 1.10 = 2.70 + 3.30 = 6.00$

£6.00
Why this scores
Mark scheme:
- M1 for $300 \times 2.5$ ( $= 750$ ) or $250 \times 2.5$ ( $= 625$ ) oe
- M1 for both $750$ and $625$
- M1 for $2 \times 1.35 + 3 \times 1.10$ ft their number of bags and pots
- A1 for $6$ or $6.00$ cao
Why this scores full marks: The candidate rounds UP to whole bags and pots — $625$ ml needs a third pot even though it is only just over two. Paying for $1.5$ bags and $2.08$ pots (about £4.32) ignores the condition in the question.
Question 8
Paper 1H/2H style — 4 marks4 marks
The ratio of Kim's age to her mother's age is $2 : 7$

In $12$ years' time, the ratio of Kim's age to her mother's age will be $2 : 5$

Work out Kim's age now. [4]
Model answer
Kim $= 2x$ , mother $= 7x$

$(2x + 12) : (7x + 12) = 2 : 5$

$5(2x + 12) = 2(7x + 12)$

$10x + 60 = 14x + 24$

$36 = 4x \quad\Rightarrow\quad x = 9$

$2 \times 9 = 18$

$18$
Why this scores
Mark scheme:
- M1 for $2x + 12$ and $7x + 12$ oe
- M1 for $5(2x + 12) = 2(7x + 12)$ oe
- M1 for $4x = 36$ oe
- A1 for $18$ cao
Why this scores full marks: Both ages increase by $12$ — the candidate adds $12$ to each before forming the equation. The answer is Kim's age ( $18$ ), not $x$ . Check: $30 : 75 = 2 : 5$ ✓.
Question 9
Paper 1H/2H style — 5 marks5 marks
Alloy A is made of copper and zinc in the ratio $3 : 2$ by mass.

Alloy B is made of copper and zinc in the ratio $1 : 4$ by mass.

Some of alloy A and some of alloy B are melted together to make $50$ kg of a new alloy in which the ratio of copper to zinc is $1 : 1$

Work out the mass of alloy A and the mass of alloy B used. [5]
Model answer
Copper is $\frac{3}{5}$ of alloy A and $\frac{1}{5}$ of alloy B.

Let the mass of A be $a$ kg, so the mass of B is $(50 - a)$ kg.

The new alloy has $25$ kg of copper.

$0.6a + 0.2(50 - a) = 25$

$0.6a + 10 - 0.2a = 25$

$0.4a = 15 \quad\Rightarrow\quad a = 37.5$

$50 - 37.5 = 12.5$

Alloy A $37.5$ kg, alloy B $12.5$ kg
Why this scores
Mark scheme:
- M1 for $\frac{3}{5}$ and $\frac{1}{5}$ (or $0.6$ and $0.2$ ) as the fractions of copper
- M1 for $0.6a + 0.2(50 - a) = 25$ oe
- M1 for $0.4a = 15$ oe
- A1 for $37.5$
- A1 for $12.5$
Why this scores full marks: The candidate turns each ratio into the FRACTION of copper and uses one unknown, so the equation is linear and short. Check: copper $22.5 + 2.5 = 25$ kg and zinc $15 + 10 = 25$ kg, so the ratio is $1 : 1$ ✓.
Question 10
Paper 1H/2H style — 5 marks5 marks
Ann and Ben have some stamps.

The ratio of the number of Ann's stamps to the number of Ben's stamps is $7 : 3$

Ann has $84$ more stamps than Ben.

Ann gives some of her stamps to Ben. The ratio is now $3 : 2$

Work out how many stamps Ann gives to Ben. [5]
Model answer
$7 - 3 = 4$ parts $= 84$

$1 \text{ part} = 84 \div 4 = 21$

$\text{Ann} = 7 \times 21 = 147, \qquad \text{Ben} = 3 \times 21 = 63$

$\text{total} = 147 + 63 = 210$

After: $3 + 2 = 5$ parts

$210 \div 5 = 42, \qquad \text{Ann} = 3 \times 42 = 126$

$147 - 126 = 21$

$21$
Why this scores
Mark scheme:
- M1 for $84 \div 4$ ( $= 21$ )
- A1 for $147$ and $63$
- M1 for $210 \div 5$ ( $= 42$ ) ft their total
- M1 for $147 - 3 \times 42$ oe ft
- A1 for $21$ cao
Why this scores full marks: The candidate spots that the TOTAL number of stamps does not change when Ann gives some to Ben, so the new ratio shares the same $210$ . Check: Ben now has $63 + 21 = 84$ , and $126 : 84 = 3 : 2$ ✓.

4MA1 Ratio and Proportion

At a glance

What you’ll learn

Working with ratios

Direct proportion

Inverse proportion

Real-life proportion — best buys, exchange rates

Quick recap

Memorise this

How it’s examined

Worked ExamplesStep-by-step solutions to past-paper-style questions

2Direct proportion problem

3Direct proportion using $y \propto x$

4Inverse proportion

5Ratio to fraction conversion

6Simplify a ratio with units

7From a fraction of a group to a ratio [1 mark]

8Direct proportion — cost of different masses [2 marks]

10Best buy across two currencies [5 marks]

Model AnswersFull mark-scheme-style answers

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

4MA1 Ratio and Proportion

At a glance

What you’ll learn

Working with ratios

Direct proportion

Inverse proportion

Real-life proportion — best buys, exchange rates

Quick recap

Memorise this

How it’s examined

Worked ExamplesStep-by-step solutions to past-paper-style questions

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

7From a fraction of a group to a ratio [1 mark]

8Direct proportion — cost of different masses [2 marks]

9Sharing when one share is a fraction of the other [3 marks]

10Best buy across two currencies [5 marks]

Model AnswersFull mark-scheme-style answers

3Direct proportion using $y \propto x$