IGCSE Maths 0580 Ratio, Proportion & Rates Notes

Overview

What you’ll learn

Simplify ratios, including those with mixed units, and divide a quantity in a given ratio.
Use proportional reasoning in recipes, map scales and best-buy comparisons.
Calculate with rates of pay, exchange rates, flow rates and fuel consumption.
Use speed = distance ÷ time, including average speed over multi-stage journeys.
Calculate density, pressure and population density from formulas given in the question.
Express direct and inverse proportion algebraically (y ∝ x, y ∝ x², y ∝ √x, y ∝ 1/x²) and find the constant of proportionality (Extended).

Syllabus checklist (0580)

Statements are paraphrased and grouped by tier. Core is studied by everyone; Extended content is examined on Papers 2 and 4 only.

Core — every candidate

1.11Understand and use ratio to give ratios in their simplest form, divide a quantity in a given ratio, and use proportional reasoning and ratios in context such as recipes, map scales and best value.
1.12Use common measures of rate, e.g. hourly rates of pay, exchange rates between currencies, flow rates and fuel consumption.
1.12Apply other measures of rate, e.g. pressure, density and population density, using formulas given in the question.
1.12Solve problems involving average speed, using speed = distance ÷ time.

Extended — Extended candidates only

E2.8Express direct and inverse proportion in algebraic terms, including linear, square, square root, cube and cube root proportion, and use the form to find unknown quantities. The symbol ∝ is required. (Extended)

In 30 seconds

1Simplify a ratio by dividing every part by the same factor: 45 : 60 = 3 : 4; convert units first (1.2 kg : 300 g = 1200 : 300 = 4 : 1).
2To divide in a ratio, add the parts, divide the total, multiply back: 360 in 5 : 4 → 9 shares → 200 and 160.
3A map scale 1 : 50 000 means 1 cm on the map is 50 000 cm (= 500 m) on the ground.
4Best value compares unit prices — price per gram, per litre or per item — not the sticker price.
5Speed = distance ÷ time; give times in a single unit first (3 h 45 min = 3.75 h). Average speed = total distance ÷ total time, never the average of the speeds.
6Density = mass ÷ volume (g/cm³ or kg/m³); pressure = force ÷ area — formulas are printed in the question when needed.
7Extended — proportion: write y = kxⁿ first, find k from one pair of values, then substitute. Inverse proportion puts the variable in the denominator: p = k/q².

Notes · 1 of 4

Ratio: simplify, split, scale

Simplifying ratios, sharing in a ratio, and ratio in recipes, maps and shops.

A ratio compares parts of the same kind: cordial to water, flour to butter. Simplify by dividing every part by a common factor: 45 : 60 = 3 : 4. Ratios need matching units first — 1.2 kg to 300 g is 1200 g : 300 g = 4 : 1. Ratios carry no units: multiplying both parts by the same number leaves the ratio unchanged, which is what makes scaling possible.

To divide a quantity in a given ratio, add the parts to get the number of shares. Sharing 360 in the ratio 5 : 4 gives 5 + 4 = 9 shares of 360 ÷ 9 = 40, so the parts are 200 and 160. Check: 200 + 160 = 360 and 200 : 160 simplifies back to 5 : 4. For three parts — 240 split in 5 : 3 : 2 — there are 10 shares of 24: 120, 72, 48.

Proportional reasoning in context. A recipe for 8 pancakes using 250 g of flour scales by the factor 14 ÷ 8 = 1.75, needing 250 × 1.75 = 437.5 g for 14 pancakes. A map scale of 1 : 50 000 turns 3.4 cm on the map into 3.4 × 50 000 = 170 000 cm = 1.7 km. A best buy compares unit prices: 750 g for 2.70 costs 36 p per 100 g, while 1.2 kg for 4.08 costs 34 p per 100 g — the larger box is better value despite the bigger sticker price.

Notes · 2 of 4

Proportion, direct and inverse

Scaling recipes by a factor, and the algebra of ∝ at Extended.

Two quantities are in direct proportion when multiplying one multiplies the other by the same factor — currency, recipes, map distances. The method: find the factor connecting the given values, then apply it. If 5 identical notebooks cost 6.75, then 12 cost 6.75 × (12 ÷ 5) = 16.20. Find one-first values (the cost of 1) if the factor is awkward.

Extended. In algebra, direct proportion reads y ∝ x, written as an equation y = kx once the constant k is fixed. Powers give other laws: y ∝ x², y ∝ √x, y ∝ x³. If y ∝ x² and y = 3 when x = 2, then 3 = k × 2², so k = 3/4 and y = 0.75x²; at x = 4, y = 0.75 × 16 = 12. Inverse proportion divides: p ∝ 1/q². If p = 20 when q = 3, then k = p q² = 180, so p = 180/q²; doubling q to 6 leaves p = 180/36 = 5.

Proportion laws at a glance (Extended)
Statement	Equation	k from (x, y) = (2, 6)
y ∝ x	y = kx	k = 3
y ∝ x²	y = kx²	k = 1.5
y ∝ √x	y = k√x	k = 6/√2 ≈ 4.24
y ∝ 1/x²	y = k/x²	k = 24

Notes · 3 of 4

Rates and speed

Pay, fuel, flow — and the speed–distance–time triangle done carefully.

A rate is one quantity per unit of another: 12.50 per hour of pay, 6.4 L per 100 km of fuel, 15 L per minute of flow. Convert the units before calculating: a car using 6.4 L per 100 km over 460 km burns 6.4 × 4.6 = 29.44 L. Flow rates multiply by time: filling a 225 L tank at 15 L/min takes 225 ÷ 15 = 15 minutes.

Speed is distance per time: speed = distance ÷ time, distance = speed × time, time = distance ÷ speed. A car covering 210 km in 2 h 30 min averages 210 ÷ 2.5 = 84 km/h. Unit conversions matter: 72 km/h = 72 000 m ÷ 3600 s = 20 m/s, and a runner covering 90 m at 7.5 m/s takes 90 ÷ 7.5 = 12 seconds.

Average speed is total distance ÷ total time for the whole journey, not the mean of the speeds. A van does 120 km at 80 km/h (1.5 h) and then 60 km at 60 km/h (1 h): total 180 km in 2.5 h, so the average speed is 180 ÷ 2.5 = 72 km/h. The plain average of the speeds, (80 + 60)/2 = 70, gives the wrong value — here 72, because more of the distance was covered at the faster speed. The total-over-total method is the one that works.

Notes · 4 of 4

Density, pressure and population density

Compound measures from formulas given in the question.

Density = mass ÷ volume, in g/cm³ or kg/m³. A metal block of mass 624 g and volume 80 cm³ has density 624 ÷ 80 = 7.8 g/cm³. Rearranged, mass = density × volume: a concrete slab of density 2500 kg/m³ and volume 0.04 m³ has mass 2500 × 0.04 = 100 kg. The syllabus prints the formula in the question when it is needed — you rearrange and substitute.

Pressure = force ÷ area, measured in N/m² (pascals). A crate exerting 720 N over a base of 0.48 m² presses with 720 ÷ 0.48 = 1500 N/m². Smaller area means bigger pressure — the reason stiletto heels dent floors. Population density divides people by area: a city of 8 400 000 people spread over 1200 km² has 8 400 000 ÷ 1200 = 7 000 people per km².

Revise

Key terms

The words examiners expect you to use precisely.

Ratio: A comparison of two or more quantities of the same kind, written with colons.45 : 60 simplifies to 3 : 4.
Share (ratio part): One equal portion when a quantity is divided in a ratio.Total ÷ (sum of parts) gives one share.
Direct proportion: A relationship where multiplying one quantity multiplies the other by the same factor.Recipes, currency, map scales.
Inverse proportion (Extended): A relationship where multiplying one quantity divides the other by the same factor.p = k/q²; speed–time for a fixed journey.
Constant of proportionality (Extended): The fixed number k in a proportion equation.Found from one known pair of values.
Unit price: The price per single unit — gram, litre or item — used to compare value.Lower unit price is the better buy.

Show 4 more key terms

Rate: One quantity per unit of another, e.g. 12.50 per hour.Pay, exchange, flow, fuel.
Average speed: Total distance divided by total time for a whole journey.Not the average of the stage speeds.
Density: Mass per unit volume, e.g. g/cm³.Density = mass ÷ volume.
Pressure: Force per unit area, in N/m² (pascals).Pressure = force ÷ area.

Revise

Common mistakes

Where students most often lose marks in this topic — and what to do instead.

Mistake: Adding the same number to both parts of a ratio.
Ratios scale by multiplication, not addition.
Instead: Only multiplying or dividing every part by the same factor keeps a ratio equivalent.
Mistake: Dividing the quantity by the number of ratio entries.
The parts are not necessarily equal; the divisor is the sum of the ratio parts.
Instead: Add the parts (5 : 4 → 9 shares), then divide and multiply back.

Show 10 more mistakes

Mistake: Simplifying a ratio with mismatched units.
1.2 kg : 300 g is not 1.2 : 300.
Instead: Convert to the same unit first: 1200 : 300 = 4 : 1.
Mistake: Comparing best buys by sticker price alone.
A cheaper label can be worse value per gram.
Instead: Compute unit prices for identical units and compare those.
Mistake: Averaging the two speeds for an average speed.
Stages cover different distances in different times.
Instead: Average speed = total distance ÷ total time, stops included.
Mistake: Entering 2 h 30 min as 2.30 hours.
Decimal hours and minutes are different scales — 0.30 h is 18 min.
Instead: Convert minutes to hours: 2 h 30 min = 2.5 h.
Mistake: Mixing kg with cm³ in a density calculation.
The units must match the g/cm³ scale.
Instead: Convert mass to grams before dividing by cm³ (or convert volume to m³ for kg/m³).
Mistake: Using one foot’s area for the whole weight.
Standing on two feet spreads the force over double the area.
Instead: Pressure = force ÷ total contact area.
Mistake: Writing inverse proportion as y = kx with a negative k.
Inverse proportion divides by the variable; it does not subtract.
Instead: y ∝ 1/x means y = k/x; y ∝ 1/x² means y = k/x².
Mistake: Reusing k from a direct-proportion law in an inverse one.
The constant is fixed by k = y/xⁿ or k = y·xⁿ depending on the law.
Instead: Build the equation first, then substitute the known pair to find k.
Mistake: Stopping at 9 : 12 when simplifying 45 : 60.
Dividing by 5 only partially cancels the ratio.
Instead: Divide by the highest common factor: 45 : 60 = 3 : 4.
Mistake: Converting km/h to m/s with only one of the two steps.
Kilometres must become metres and “per hour” must become “per second”.
Instead: Multiply by 1000 and divide by 3600 — in one step, divide by 3.6.

Check yourself

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Question 1 of 3

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Simplify a ratio

What is 45 : 60 in its simplest form?

Practise

Topical questions

14 exam-style questions with model answers and mark schemes.

A set of structured and calculation questions matched to this topic, in three tiers, each with a model answer, mark scheme and examiner tip.

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Exam practice

Practice paper

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Question paper

Topic 3 Practice Paper

30 marks · 40 min · 5 pages

10 multiple-choice questions and structured questions with answer lines. Topic 3: simplifying ratios, dividing quantities in a ratio, recipes, map scales and best buys, direct and inverse proportion, speed and average speed, density, pressure, population density, pay and fuel rates.

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Mark scheme

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Topic 3 Practice Paper — mark scheme

Answers · Marking points · 3 pages

Model answers with a mark for each creditworthy point — for marking at home or in class.

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Which papers examine this topic?

Paper 1 / 3

Core: 80 marks each, 1 h 30 each

Paper 1 non-calculator, Paper 3 calculator

Paper 2 / 4

Extended: 100 marks each, 2 h each

Paper 2 non-calculator, Paper 4 calculator

Accuracy

3 s.f., angles to 1 d.p.

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Questions

Frequently asked questions

How do I know whether to add or multiply when sharing in a ratio?

Sharing is always: add the ratio parts to get the number of shares, divide the total by that sum, then multiply by each part. Ratios themselves only ever scale by multiplying or dividing both sides by the same number.

Why is average speed not the average of the speeds?

Because the stages take different times. Average speed is total distance over total time. A 20-minute rest or a longer slow stage drags the average down in a way the simple mean cannot see.

What does “inversely proportional to the square” actually mean?

It means the quantity equals a constant divided by the distance squared: B = k/d². Double the distance and B falls to a quarter. Fix k from one known pair, then substitute the new value.

Do I need to memorise the density and pressure formulas?

No — the syllabus says the required formulas will be given in the question. You do need to rearrange them, substitute carefully, and keep the units consistent (grams with cm³, kilograms with m³).

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IGCSE Maths Ratio, proportion, rates notes