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

In Cambridge IGCSE Mathematics, a ratio is a way to compare two quantities by showing the relative size of one quantity to another, often expressed as 'a:b'. A proportion, on the other hand, is an equation that states that two ratios are equal. Understanding the difference is crucial for solving problems involving comparisons and scaling.

How do you simplify a ratio in Mathematics?

To simplify a ratio in Mathematics, divide both terms 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. This process helps in making comparisons easier and is a fundamental skill in the Cambridge IGCSE curriculum.

How can I solve a proportion problem using cross-multiplication?

To solve a proportion problem using cross-multiplication, set up the proportion as an equation of two ratios, such as a/b = c/d. Then, cross-multiply by multiplying a by d and b by c, resulting in the equation ad = bc. Solve for the unknown variable by isolating it on one side of the equation. This method is commonly used in Cambridge IGCSE Mathematics to find missing values in proportional relationships.

What are some real-life applications of ratios and proportions?

Ratios and proportions are widely used in real-life applications such as cooking, where recipes require ingredients in specific ratios, and in map reading, where scales represent proportional distances. They are also essential in financial contexts, such as calculating interest rates or comparing prices. Understanding these concepts is vital for practical problem-solving, as emphasized in the Cambridge IGCSE Mathematics curriculum.

How do you convert a ratio to a fraction in Mathematics?

To convert a ratio to a fraction in Mathematics, take the two terms of the ratio, a:b, and express it as a fraction a/b. This conversion is useful for performing arithmetic operations and solving equations involving ratios. Mastery of this skill is important for students studying Cambridge IGCSE Mathematics, as it aids in understanding and manipulating proportional relationships.

Why is "Treating an inverse proportion question as direct" a common mistake in Ratios and Proportions?

Why it happens: The phrase "inversely proportional" is missed and students set up y = kx. How to avoid it: Highlight the words "inversely" or "directly" before writing any equation. Source: 0580/42 May/Jun 2024 — examiner report Q9

Why is "Forgetting to find the constant of proportionality" a common mistake in Ratios and Proportions?

Why it happens: Students substitute the new x into the proportion symbol expression directly without solving for k first. How to avoid it: Step 1 is **always** to find k from the given pair, **then** substitute the new value.

Why is "Leaving a ratio unsimplified when the question asks for simplest form" a common mistake in Ratios and Proportions?

Why it happens: Students stop after solving without checking the final instruction. How to avoid it: Always divide all parts of the ratio by their HCF.

Why is "Missing the "square" in "proportional to the square of"" a common mistake in Ratios and Proportions?

Why it happens: Quick reading of the question. How to avoid it: Underline the modifier ("square", "cube", "square root") in the question before writing y = k/x^n. Source: 0580 Extended examiner reports — recurring

Cambridge IGCSE Mathematics (0580)

Ratios and Proportions

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Short Notes - Ratios and Proportions

Detailed Study Notes

Detailed notes on Number for Cambridge IGCSE Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Ratios and Proportions — Cambridge IGCSE 0580 Maths Extended (2026)

Ratios scale, simplify and split. Proportion tells you whether two quantities scale together or against each other. The two ideas underpin scale drawings, recipes, currency conversion, mixtures and similarity.

At a glance

A ratio compares quantities. $a : b$ means " $a$ parts to $b$ parts".
Simplify a ratio by dividing by the HCF: $12 : 18 = 2 : 3$ .
Sharing in a ratio: total $\div$ (sum of parts) gives the value of one part.
Direct proportion: $y \propto x$ → $y = kx$ . As $x$ goes up, $y$ goes up.
Inverse proportion: $y \propto \dfrac{1}{x}$ → $y = \dfrac{k}{x}$ . As $x$ goes up, $y$ goes DOWN.
Find $k$ by substituting one known $(x, y)$ pair, then use the formula for any other case.
Scale drawings: '1 : 50' means 1 cm on paper = 50 cm in reality.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E1.12 — Use ratio to compare quantities; simplify ratios; divide a quantity in a given ratio.
E1.14 — Use direct and inverse proportion; recognise the language and notation $y \propto x$ and $y \propto 1/x$ .
E1.15 — Use scale on maps and drawings.

Reading, simplifying and using ratios

A ratio shares the SAME UNITS. Simplify by dividing through; expand to share a total.

A ratio $a : b$ compares two amounts of the SAME unit. Reading: " $a$ parts to $b$ parts".

Simplify by dividing both sides by their HCF.

$12 : 18$ → divide by $6$ → $2 : 3$ .
$25 : 35 : 50$ → divide by $5$ → $5 : 7 : 10$ .

If the ratio uses different units, convert them to the SAME unit first.

$300\,\text{g} : 1.2\,\text{kg}$ → convert: $300\,\text{g} : 1200\,\text{g} = 1 : 4$ .

Sharing in a ratio. Add the parts; that's your divisor.

Worked. Share $240 between $A$ and $B$ in the ratio $5 : 7$ .

Total parts: $5 + 7 = 12$ .
One part: $\dfrac{240}{12} = 20$ , so each part is $20.
$A$ gets $5 \times 20 = 100$ , so $100. $B$ gets $7 \times 20 = 140$ , so $140.
Check: $100 + $140 = $240. ✓

Equivalent ratios. $1 : 4$ is the same as $2 : 8$ is the same as $5 : 20$ . Multiply or divide both sides by the same non-zero number.

Add the parts (5 + 7 = 12), find one part, then multiply out each share.

Same UNITS first, then simplify.
Divide both sides by the HCF to simplify fully.
Sharing a total: total $\div$ (sum of parts) = one part.
Always check the parts add back to the total.

Direct proportion

Two quantities go up together at the same rate. Common in recipes, exchange rates and unit pricing.

If $y$ is directly proportional to $x$ , written $y \propto x$ , then there is a constant $k$ so that $y = k x.$

The constant $k$ (the constant of proportionality) is the ratio $y/x$ — the same for every valid pair.

Three-step method.

Write $y = k x$ .
Substitute the known $(x, y)$ to find $k$ .
Use the formula for the new $x$ or $y$ .

Worked. " $y$ is directly proportional to $x$ . When $x = 6$ , $y = 15$ . Find $y$ when $x = 10$ ."

$y = kx$ .
$15 = k \times 6 \Rightarrow k = 2.5$ .
$y = 2.5 \times 10 = 25$ .

Find k from the known pair, then use y = kx for the new value.

Real contexts.

Recipe scaling. " $3$ apples make $300\,\text{g}$ of sauce. How much for $5$ apples?" → $k = 100\,\text{g/apple}$ → $5 \times 100 = 500\,\text{g}$ .
Currency. "$1 = £0.78. Convert $245." → $245 \times 0.78 = 191.10$ , so £191.10.

Other forms. Cambridge can write direct proportion as $y \propto x^{2}$ , $y \propto \sqrt{x}$ , etc. Same method — just the formula changes.

$y \propto x^{2}$ : $y = k x^{2}$ .
$y \propto \sqrt{x}$ : $y = k \sqrt{x}$ .

$y \propto x$ means $y = kx$ for some constant $k$ .
Find $k$ from a single data point. Then use it.
Up together: as $x$ doubles, $y$ doubles.
Variants: $y \propto x^{2}$ , $y \propto \sqrt{x}$ — same $y = k \cdot (\text{stuff})$ shape.

Inverse proportion

When one goes up, the other goes down. Workers and time, speed and journey time, pressure and volume.

If $y$ is inversely proportional to $x$ , written $y \propto \dfrac{1}{x}$ , then there is a constant $k$ so that $y = \dfrac{k}{x} \quad \text{equivalently} \quad x y = k.$

The product $x y$ is constant — that's the test.

Method.

Write $xy = k$ (or $y = k/x$ ).
Substitute the known $(x, y)$ to find $k$ .
Use it for the new pair.

Worked. " $y$ is inversely proportional to $x$ . When $x = 4$ , $y = 9$ . Find $y$ when $x = 12$ ."

$xy = k$ → $4 \times 9 = k = 36$ .
New $x = 12$ → $12 \times y = 36 \Rightarrow y = 3$ .

As x triples (4 → 12), y is divided by 3 (9 → 3), so the product 36 never changes.

Real contexts.

Workers and time. "It takes $6$ workers $8$ hours to lay a path. How long with $4$ workers (same rate)?" Inverse: $6 \times 8 = 4 \times t \Rightarrow t = 12\,\text{h}$ .
Speed and time. "A trip at $40\,\text{km/h}$ takes $3$ hours. At $60\,\text{km/h}$ ?" Inverse: $40 \times 3 = 60 \times t \Rightarrow t = 2\,\text{h}$ .

Variants. $y \propto \dfrac{1}{x^{2}}$ → $y = \dfrac{k}{x^{2}}$ . (Common in physics: gravity, light intensity.)

Direct proportion plots as a line through the origin; inverse proportion as a falling curve.

$y \propto 1/x$ means $xy = k$ .
Up one, down the other.
Test: is the product constant across pairs?
Variants: $y \propto 1/x^{2}$ (inverse-square law).

Scale on maps and drawings

A scale is a ratio of distances. Convert by multiplying or dividing — and always sanity-check the units.

A scale of "1 : 50" means $1\,\text{cm}$ on the drawing represents $50\,\text{cm}$ in reality. The ratio uses the SAME units on both sides.

Drawing → real distance. Multiply the drawing distance by the scale factor.

Scale $1 : 200$ , drawing distance $4\,\text{cm}$ → real $= 4 \times 200 = 800\,\text{cm} = 8\,\text{m}$ .

Drawing → real: multiply by the scale factor (the bars are schematic, not drawn to scale).

Real → drawing distance. Divide the real distance by the scale factor.

Scale $1 : 200$ , real distance $15\,\text{m} = 1500\,\text{cm}$ → drawing $= 1500 / 200 = 7.5\,\text{cm}$ .

Map scales are often "1 : 50,000" (a typical OS map). $1\,\text{cm}$ on the map = $50{,}000\,\text{cm} = 500\,\text{m}$ .

Two-step scaled distances. Question. "On a map of scale $1 : 25{,}000$ , two villages are $8\,\text{cm}$ apart. What's the real distance in km?"

$8\,\text{cm} \times 25{,}000 = 200{,}000\,\text{cm} = 2{,}000\,\text{m} = 2\,\text{km}$ .

$1 : n$ means 1 unit on paper = $n$ units in reality.
Drawing → real: multiply.
Real → drawing: divide.
Always reduce to the unit the question asks for (cm, m, km).

Quick recap

Ratios share the same units. Simplify by dividing both sides by the HCF.
Sharing in a ratio: total $\div$ (sum of parts) gives one part.
$y \propto x$ → $y = kx$ (direct, both grow). $y \propto 1/x$ → $xy = k$ (inverse, one grows when the other shrinks).
Find $k$ from one data point; use it for everything else.
Scale $1 : n$ : paper × $n$ = reality.
Length scale $k$ → area scale $k^{2}$ , volume scale $k^{3}$ .

Memorise this

Verbatim phrases and definitions Cambridge mark schemes credit.

Ratio — a comparison of two or more quantities measured in the same units, written $a : b$ .
Direct proportion — when $y = kx$ for some constant $k$ ; $y$ and $x$ change in the same direction.
Inverse proportion — when $xy = k$ for some constant $k$ ; $y$ and $x$ change in opposite directions.
Constant of proportionality ( $k$ ) — the constant that links two proportional quantities.
Scale factor — the multiplier between corresponding lengths in two similar figures or between drawing and real measurements.

How it’s examined

Ratio splitting ("share an amount in the ratio a : b : c") is a recurring 2-3 mark Paper 2 question. Direct- and inverse-proportion questions ask you to find the constant of proportionality from one pair and use it on a second — often dressed as recipe-scaling, fuel-consumption or workforce problems on Paper 4. Scale-and-map questions are usually a 2-mark item with a unit-conversion sting. Examiner reports flag failure to convert units before simplifying ratios as a top recurring slip.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E1.12, E1.14, E1.15); 0580/22 May/Jun 2024 — Q6 (sharing in a ratio); 0580/42 Oct/Nov 2024 — Q9 (inverse proportion); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

Take this whole topic with you

Step-by-step worked examples — Ratios and Proportions

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

Question type:

Question patterns to master — Ratios and Proportions

Almost every ratios and proportions exam question is one of these shapes. Learn to spot each one and you will always know how to start.

Direct calculation

Recognise it by

A single instruction — share in the ratio, find $y$ when …, simplify the ratio — with the relationship ( $y \propto x^{n}$ or a ratio) already given.

How to approach it

For sharing, total the parts and find one part. For proportion, write $y=kx^{n}$ (or $\frac{k}{x^{n}}$ ), find $k$ from the given pair, then substitute the new value.

Common trap

Examiner reports flag substituting the new value before finding $k$ , and missing the modifier — 'square', 'cube', 'square root' — so the power $n$ is wrong.

Word problem

Recognise it by

A real-world context — a recipe, a map scale, an unequal share — where you must set up the ratio or proportion yourself.

How to approach it

Translate the context into a ratio or an equation in a common multiplier $x$ , converting all quantities to a single unit first, then solve and answer in the unit requested.

Common trap

Examiner reports flag skipping unit conversion before simplifying, and setting one share equal to a comparative difference rather than to $5x-2x$ .

Multi-step problem

Recognise it by

Several pieces chained together — a ratio that changes after an event, or a share subject to an extra condition such as a minimum total.

How to approach it

Introduce a variable $x$ for the common multiplier, express every quantity in terms of it, write each condition as an equation, then solve.

Common trap

Examiner reports flag candidates who never translate the ratio into algebraic parts, leaving no equation to solve.

Show that / prove

Recognise it by

An instruction to justify with calculations or decide which option is best — the supporting comparison is required, not just the choice.

How to approach it

Compute the comparable quantity (price per gram, unit rate) for every option and state explicitly which wins and why.

Common trap

Examiner reports flag a bare answer with no comparison — all calculations must be shown for full marks.

1Share an amount in a given ratio
CoreDirect calculation• Adapted from 0580/22 Oct/Nov 2023 Q4• ratio, sharing
Question
Share $420 in the ratio $3 : 4 : 7$ .
Step-by-step solution
1. Step 1
  Total parts: $3 + 4 + 7 = 14$ .
2. Step 2
  One part: $\dfrac{420}{14} = 30$ .
3. Step 3
  Multiply: $3 \times 30 = 90$ , $4 \times 30 = 120$ , $7 \times 30 = 210$ — that is $90, $120 and $210.
4. Step 4
  Check: $90 + 120 + 210 = 420$ . ✓
Answer
$90 : $120 : $210
Examiner tip
Always verify your three values sum to the original amount — examiners deduct only when this check fails.
2Solve a direct proportion problem
CoreDirect calculation• direct proportion
Question
$y$ is directly proportional to $x$ . When $x = 6$ , $y = 27$ . Find $y$ when $x = 14$ .
Step-by-step solution
1. Step 1
  Direct proportion gives $y = kx$ . Find $k$ .
  $27 = k \times 6 \Rightarrow k = 4.5$
2. Step 2
  Substitute $x = 14$ .
  $y = 4.5 \times 14 = 63$
Answer
$y = 63$
3Solve an inverse proportion problem
ExtendedDirect calculation• Adapted from 0580/42 May/Jun 2024 Q9• inverse proportion
Question
$y$ is inversely proportional to the square of $x$ . When $x = 2$ , $y = 25$ . Find $y$ when $x = 5$ .
Step-by-step solution
1. Step 1
  $y = \dfrac{k}{x^{2}}$ .
2. Step 2
  Find $k$ : $25 = \dfrac{k}{4} \Rightarrow k = 100$ .
3. Step 3
  Substitute $x = 5$ : $y = \dfrac{100}{25} = 4$ .
Answer
$y = 4$
Examiner tip
Inverse-of-square is a frequent extension twist. Read carefully: "inversely proportional to the square of $x$ " gives $\dfrac{k}{x^{2}}$ , not $\dfrac{k}{x}$ .
4Simplify and use equivalent ratios
CoreWord problem• equivalent ratios, simplifying
Question
A recipe uses flour and sugar in the ratio $5 : 2$ . If $750\,\text{g}$ of flour is used, how much sugar is needed?
Step-by-step solution
1. Step 1
  Set up: $\dfrac{\text{flour}}{\text{sugar}} = \dfrac{5}{2}$ .
2. Step 2
  Cross-multiply: $5 \times \text{sugar} = 2 \times 750 \Rightarrow \text{sugar} = \dfrac{1500}{5} = 300\,\text{g}$ .
Answer
$300\,\text{g}$ of sugar
5Simplify a ratio that contains different units
ExtendedDirect calculation• Adapted from 0580/22 May/Jun 2023 Q6• ratio, units, simplifying
Question
Write the ratio $45\,\text{cm} : 1.2\,\text{m} : 750\,\text{mm}$ in its simplest form.
Step-by-step solution
1. Step 1
  Choose a single unit. Millimetres is easiest: $45\,\text{cm} = 450\,\text{mm}$ ; $1.2\,\text{m} = 1200\,\text{mm}$ ; $750\,\text{mm}$ stays.
2. Step 2
  Ratio becomes $450 : 1200 : 750$ .
3. Step 3
  Divide every part by their HCF. $\gcd(450,1200,750) = 150$ .
  $\dfrac{450}{150} : \dfrac{1200}{150} : \dfrac{750}{150} = 3 : 8 : 5$
Answer
$3 : 8 : 5$
Examiner tip
The 2023 examiner report flags that candidates who skip the unit-conversion step almost always lose all three method marks. Convert to a single unit before you simplify.
6Given one share of a ratio, find the total amount
ExtendedWord problem• Adapted from 0580/42 Oct/Nov 2022 Q3(a)• ratio, sharing, reverse
Question
Amir, Bina and Chen share an amount of money in the ratio $2 : 5 : 7$ . Bina receives $155 more than Amir. Find the total amount shared.
Step-by-step solution
1. Step 1
  Let one part equal $x$ dollars. Amir gets $2x$ , Bina gets $5x$ , Chen gets $7x$ .
2. Step 2
  Bina's share exceeds Amir's by $5x - 2x = 3x$ .
  $3x = 155 \Rightarrow x = \dfrac{155}{3}$
3. Step 3
  Total parts $= 2 + 5 + 7 = 14$ , so total amount $= 14x$ .
  $14x = 14 \times \dfrac{155}{3} = \dfrac{2170}{3} \approx 723.33$
Answer
$723.33 (to the nearest cent)
Examiner tip
The examiner report flags that candidates often set $5x = 155$ instead of identifying the difference of two shares. Underline the comparative phrase ("more than", "twice as much") and translate to an algebraic equation in $x$ .
7Direct proportion to the square of $x$
ExtendedDirect calculation• Adapted from 0580/42 Feb/Mar 2024 Q7• direct proportion, squared
Question
$y$ is directly proportional to the square of $x$ . When $x = 3$ , $y = 18$ . (a) Find an expression for $y$ in terms of $x$ . (b) Find $y$ when $x = 7$ .
Step-by-step solution
1. Step 1
  Write the proportion equation.
  $y = kx^{2}$
2. Step 2
  Substitute $x = 3$ , $y = 18$ to find $k$ : $18 = k \times 9 \Rightarrow k = 2$ .
3. Step 3
  (a) Equation: $y = 2x^{2}$ .
  $y = 2x^{2}$
4. Step 4
  (b) Substitute $x = 7$ : $y = 2 \times 49 = 98$ .
Answer
(a) $y = 2x^{2}$ (b) $y = 98$
Examiner tip
Examiners reward candidates who write the equation $y = kx^{2}$ first — this earns the method mark even if the value of $k$ is wrong. Skipping straight to the answer loses 2 of 3 marks.
8Direct proportion to the cube of $x$
ExtendedDirect calculation• Adapted from 0580/42 May/Jun 2023 Q10• direct proportion, cubed
Question
The volume $V$ of a sphere is directly proportional to the cube of its radius $r$ . When $r = 2\,\text{cm}$ , $V = 33.5\,\text{cm}^{3}$ . Find $V$ when $r = 6\,\text{cm}$ .
Step-by-step solution
1. Step 1
  Write $V = kr^{3}$ and find $k$ .
  $33.5 = k \times 8 \Rightarrow k = 4.1875$
2. Step 2
  Substitute $r = 6$ : $V = 4.1875 \times 216 = 904.5\,\text{cm}^{3}$ .
3. Step 3
  Shortcut to spot: tripling $r$ multiplies $V$ by $3^{3} = 27$ , so $V = 33.5 \times 27 = 904.5$ . ✓
Answer
$V = 904.5\,\text{cm}^{3}$
Examiner tip
The 2023 mark scheme awards a method mark for the scale-factor shortcut ( $k$ -free) as well as the algebraic route. Either is acceptable — just be explicit about which you use.
9Use a map scale to find a real distance
ExtendedWord problem• Adapted from 0580/22 Oct/Nov 2022 Q10• scale, map
Question
A map is drawn to a scale of $1 : 25{,}000$ . Two villages are $14.6\,\text{cm}$ apart on the map. Find the real distance between the villages in kilometres.
Step-by-step solution
1. Step 1
  Scale $1 : 25{,}000$ means $1\,\text{cm}$ on the map represents $25{,}000\,\text{cm}$ in real life.
2. Step 2
  Real distance in cm: $14.6 \times 25{,}000 = 365{,}000\,\text{cm}$ .
3. Step 3
  Convert to km: $365{,}000\,\text{cm} = 3650\,\text{m} = 3.65\,\text{km}$ .
  $365{,}000\,\text{cm} \div 100{,}000 = 3.65\,\text{km}$
Answer
$3.65\,\text{km}$
Examiner tip
The examiner report flags that candidates routinely state the answer in cm or m. Always read the unit demanded by the question — converting to km is worth the final accuracy mark.
10Inverse proportion to the cube root of $x$
ExtendedDirect calculation• Adapted from 0580/42 Oct/Nov 2024 Q11• inverse proportion, cube root
Question
$y$ is inversely proportional to $\sqrt[3]{x}$ . When $x = 8$ , $y = 12$ . Find $y$ when $x = 64$ .
Step-by-step solution
1. Step 1
  Write the proportion equation.
  $y = \dfrac{k}{\sqrt[3]{x}}$
2. Step 2
  Substitute $x = 8$ (so $\sqrt[3]{8} = 2$ ): $12 = \dfrac{k}{2} \Rightarrow k = 24$ .
3. Step 3
  Substitute $x = 64$ (so $\sqrt[3]{64} = 4$ ): $y = \dfrac{24}{4} = 6$ .
Answer
$y = 6$
Examiner tip
Cube-root inverse proportion is a recurring A* twist. The 2024 mark scheme awards method marks for the $y = \dfrac{k}{\sqrt[3]{x}}$ line and for finding $k$ , even if the final substitution slips.
11Decrease a quantity in a given ratio
ExtendedDirect calculation• ratio, decrease
Question
A photograph $24\,\text{cm}$ wide is reduced in the ratio $3 : 8$ (new : old). Find the width of the reduced photograph.
Step-by-step solution
1. Step 1
  Reducing in the ratio $3 : 8$ means the new width is $\dfrac{3}{8}$ of the old.
2. Step 2
  Compute: $\dfrac{3}{8} \times 24 = 9\,\text{cm}$ .
  $\text{new width} = \dfrac{3}{8} \times 24 = 9\,\text{cm}$
Answer
$9\,\text{cm}$
Examiner tip
Read carefully whether the ratio is given new:old or old:new. The examiner report repeatedly flags reversed multiplications here — write the fraction down before computing.
12Find a new ratio after a change
ExtendedMulti-step problem• Adapted from 0580/42 May/Jun 2024 Q6• ratio, multi-step
Question
Boys and girls in a class are in the ratio $3 : 2$ . If $6$ more girls join, the new ratio becomes $1 : 1$ . Find the original number of boys and girls in the class.
Step-by-step solution
1. Step 1
  Let the original counts be $3x$ boys and $2x$ girls.
2. Step 2
  After the change: still $3x$ boys but $(2x + 6)$ girls.
3. Step 3
  The new ratio $1 : 1$ means boys = girls.
  $3x = 2x + 6$
4. Step 4
  Solve: $x = 6$ .
5. Step 5
  Original boys $= 3x = 18$ ; original girls $= 2x = 12$ . Check: after $6$ more girls, $18 : 18 = 1 : 1$ ✓.
Answer
Originally $18$ boys and $12$ girls.
Examiner tip
The 2024 examiner report flags that candidates often forget to translate ratios into algebraic parts. Always introduce a variable $x$ for the common multiplier — without it, the equation cannot be set up.
13Best-buy comparison with three pack sizes
ChallengeShow that / prove• Adapted from 0580/42 Oct/Nov 2023 Q4• unit price, best buy, stretch
Question
Coffee is sold in three sizes: Small $200\,\text{g}$ for $5.40, Medium $350\,\text{g}$ for $8.75, Large $500\,\text{g}$ for $13.00. Which size offers the best value per gram? Justify with calculations.
Step-by-step solution
1. Step 1
  Compute price per gram for each pack.
  $\text{Small: } \dfrac{5.40}{200} = 0.027\,\text{\$/g}$
2. Step 2
  Medium.
  $\dfrac{8.75}{350} = 0.025\,\text{\$/g}$
3. Step 3
  Large.
  $\dfrac{13.00}{500} = 0.026\,\text{\$/g}$
4. Step 4
  Medium has the lowest price per gram ($0.025/g), so Medium is the best value.
Answer
Medium ( $350\,\text{g}$ ) at $0.025/g is the best value (Small $0.027/g, Large $0.026/g).
Examiner tip
The 2023 mark scheme demands all three calculations be shown for full marks. A bare "Medium" with no supporting comparison earns zero — examiners cannot infer your reasoning from the answer alone.
14Joint proportion — $y$ proportional to one variable and inversely to another
ChallengeDirect calculation• Adapted from 0580/42 May/Jun 2024 Q12• joint proportion, stretch
Question
$y$ is directly proportional to $x$ and inversely proportional to $z^{2}$ . When $x = 6$ and $z = 2$ , $y = 9$ . Find $y$ when $x = 10$ and $z = 5$ .
Step-by-step solution
1. Step 1
  Combine the proportions into one equation.
  $y = \dfrac{kx}{z^{2}}$
2. Step 2
  Substitute the first pair: $9 = \dfrac{k \times 6}{4} \Rightarrow 9 = \dfrac{3k}{2} \Rightarrow k = 6$ .
3. Step 3
  Substitute $x = 10$ , $z = 5$ .
  $y = \dfrac{6 \times 10}{25} = \dfrac{60}{25} = 2.4$
Answer
$y = 2.4$
Examiner tip
Joint-proportion problems are a frequent A* discriminator. The examiner report flags that candidates often write two separate proportionality statements — combine them into a single equation $y = \dfrac{kx}{z^{n}}$ from the outset.
15Three-way share with a minimum-amount condition
ChallengeMulti-step problem• Adapted from 0580/42 Oct/Nov 2024 Q9• ratio, sharing, stretch
Question
A sum of money is shared between $A$ , $B$ and $C$ in the ratio $4 : 7 : 9$ . The total is at least $1{,}200 and $C$ 's share is a whole number of dollars. Find the smallest possible total and the corresponding three shares.
Step-by-step solution
1. Step 1
  Let one part equal $x$ . Total = $20x$ . Shares: $A = 4x$ , $B = 7x$ , $C = 9x$ .
2. Step 2
  For $C$ 's share ( $9x$ ) to be a whole number of dollars, $9x \in \mathbb{Z}$ — so $x$ may be a fraction with denominator a divisor of $9$ .
3. Step 3
  Total $\geq 1200 \Rightarrow 20x \geq 1200 \Rightarrow x \geq 60$ .
4. Step 4
  The smallest $x$ with $9x \in \mathbb{Z}$ and $x \geq 60$ is $x = 60$ itself. Total $= 1200$ ; $A = 240$ , $B = 420$ , $C = 540$ .
5. Step 5
  Check: $240 + 420 + 540 = 1200$ ✓; $C = 540$ is whole ✓.
Answer
Smallest total: $1{,}200; shares $A =$ $240, $B =$ $420, $C =$ $540.
Examiner tip
The 2024 mark scheme awards a stretch mark for explicitly justifying the choice of $x$ (the smallest feasible value satisfying both conditions). State the constraints separately, then combine.

Key Formulae — Ratios and Proportions

The formulae you need to memorise for ratios and proportions on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Direct proportion
$y \propto x \iff y = kx$
$k$
the constant of proportionality (must be found from given data)
When to use
Whenever doubling $x$ doubles $y$ .
Inverse proportion
$y \propto \dfrac{1}{x} \iff y = \dfrac{k}{x}$
$k$
constant of proportionality
When to use
When increasing $x$ decreases $y$ in a fixed-product way.
Proportion to a power
$y \propto x^{n} \iff y = kx^{n}$
$n$
the given power (e.g. $\sqrt{x}$ means $n=\tfrac{1}{2}$ )
When to use
Direct or inverse proportion problems involving squares, cubes or roots.

Key Definitions and Keywords — Ratios and Proportions

Definitions to memorise and the exact keywords mark schemes credit for ratios and proportions answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Ratio
Examiner keyword
A comparison of two or more quantities of the same kind, written with colons (e.g. $3 : 4$ ).
Proportion
A statement that two ratios are equal.
Example
$\dfrac{2}{3} = \dfrac{8}{12}$ is a proportion.
Direct proportion
Examiner keyword
Two quantities are in direct proportion when their ratio is constant — as one increases, the other increases at the same rate.
Inverse proportion
Examiner keyword
Two quantities are inversely proportional when their product is constant — as one increases, the other decreases proportionally.
Constant of proportionality ( $k$ )
The fixed multiplier linking proportional quantities.
Unitary method
Find the value of one unit, then scale up.
Example
If $5$ books cost $30, one book costs $6, so $8$ books cost $48.

Common Mistakes and Misconceptions — Ratios and Proportions

The traps other students keep falling into on ratios and proportions questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Treating an inverse proportion question as direct
0580/42 May/Jun 2024 — examiner report Q9
Why it happens
The phrase "inversely proportional" is missed and students set up $y = kx$ .
How to avoid it
Highlight the words "inversely" or "directly" before writing any equation.
✕Forgetting to find the constant of proportionality
Why it happens
Students substitute the new $x$ into the proportion symbol expression directly without solving for $k$ first.
How to avoid it
Step 1 is always to find $k$ from the given pair, then substitute the new value.
✕Leaving a ratio unsimplified when the question asks for simplest form
Why it happens
Students stop after solving without checking the final instruction.
How to avoid it
Always divide all parts of the ratio by their HCF.
✕Missing the "square" in "proportional to the square of"
0580 Extended examiner reports — recurring
Why it happens
Quick reading of the question.
How to avoid it
Underline the modifier ("square", "cube", "square root") in the question before writing $y = \dfrac{k}{x^{n}}$ .

Ratios and Proportions — frequently asked questions

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

Ratios and Proportions

Short Notes - Ratios and Proportions

Detailed Study Notes

At a glance

What you’ll learn

Reading, simplifying and using ratios

Direct proportion

Inverse proportion

Scale on maps and drawings

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

Question patterns to master — Ratios and Proportions

1Share an amount in a given ratio

2Solve a direct proportion problem

3Solve an inverse proportion problem

4Simplify and use equivalent ratios

5Simplify a ratio that contains different units

6Given one share of a ratio, find the total amount

7Direct proportion to the square of xxx

8Direct proportion to the cube of xxx

9Use a map scale to find a real distance

10Inverse proportion to the cube root of xxx

11Decrease a quantity in a given ratio

12Find a new ratio after a change

13Best-buy comparison with three pack sizes

14Joint proportion — yyy proportional to one variable and inversely to another

15Three-way share with a minimum-amount condition

Direct proportion

Inverse proportion

Proportion to a power

Ratio

Proportion

Direct proportion

Inverse proportion

Constant of proportionality (kkk)

Unitary method

✕Treating an inverse proportion question as direct

✕Forgetting to find the constant of proportionality

✕Leaving a ratio unsimplified when the question asks for simplest form

✕Missing the "square" in "proportional to the square of"

Ratios and Proportions — frequently asked questions

7Direct proportion to the square of $x$

8Direct proportion to the cube of $x$

10Inverse proportion to the cube root of $x$

14Joint proportion — $y$ proportional to one variable and inversely to another

Constant of proportionality ( $k$ )