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

NumberCambridge IGCSE Mathematics (0580)

Ratios and Proportions

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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.

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.

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 $x = 10$ ."

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

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 $x = 12$ → $12 \times y = 36 \Rightarrow y = 3$ .

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.)

$y \propto 1/x$ 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 $= 4 \times 200 = 800\,\text{cm} = 8\,\text{m}$ .

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).

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.

Master this Topic

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

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

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.

1Share an amount in a given ratio
Core• 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
Core• 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
Extended• 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
Core• 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

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

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.

Ratios and Proportions — frequently asked questions

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

NumberCambridge IGCSE Mathematics (0580)

Ratios and Proportions

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

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

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.

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 $x = 10$ ."

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

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 $x = 12$ → $12 \times y = 36 \Rightarrow y = 3$ .

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.)

$y \propto 1/x$ 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 $= 4 \times 200 = 800\,\text{cm} = 8\,\text{m}$ .

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).

How it’s examined

Master this Topic

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

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

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.

1Share an amount in a given ratio
Core• 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
Core• 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
Extended• 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
Core• 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

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

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.

Ratios and Proportions — frequently asked questions

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

Ratios and Proportions

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Share an amount in a given ratio

2Solve a direct proportion problem

3Solve an inverse proportion problem

4Simplify and use equivalent ratios

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"

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Ratios and Proportions — frequently asked questions

Ratios and Proportions

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Share an amount in a given ratio

2Solve a direct proportion problem

3Solve an inverse proportion problem

4Simplify and use equivalent ratios

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"

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Ratios and Proportions — frequently asked questions

Constant of proportionality ( $k$ )

Constant of proportionality ( $k$ )