IGCSE Maths · Number · Core & Extended

Ratio, Proportion and Rates: Comparing Quantities Fairly

This topic is identical for Core and Extended — every idea here is examined the same way at both tiers. Ratio, proportion and rates all answer the same underlying question: how does one quantity compare to, or change with, another? Once you can answer that cleanly, you can handle recipes, maps, currency exchange, speed, and best-value shopping with the same small toolkit.

Why this matters Every time you scale a recipe up for guests, check which supermarket pack is actually cheaper per gram, work out how long a journey will take at a given speed, or convert your allowance into a different currency before a trip — you’re using ratio, proportion, or rate. This is one of the most immediately useful topics in the whole syllabus.

1. Understanding and Simplifying Ratios

A ratio compares two or more quantities of the same kind, written with a colon. Just like fractions, ratios should be given in their simplest form — divide every part by their HCF.

a : b : c
Worked example Simplify the ratio 20 : 30 : 40.
HCF of 20, 30 and 40 is 10. Divide every part by 10.
Answer: 2 : 3 : 4
Worked example — ratios with different units Write 750 g : 2 kg as a ratio in its simplest form.
Convert to the same unit first: 2 kg = 2000 g.
750 : 2000 → HCF is 250 → 3 : 8
Try it — Simplifying Ratios
  1. Simplify 15 : 25.
  2. Simplify 18 : 24 : 30.
  3. Write 400 m : 1 km as a ratio in simplest form.
  4. Write 45 minutes : 2 hours as a ratio in simplest form.
Show answers
  1. 3 : 5
  2. 3 : 4 : 5
  3. 400 : 1000 = 2 : 5
  4. 45 : 120 = 3 : 8

2. Dividing a Quantity in a Given Ratio

Add up the parts of the ratio to find the total number of “shares,” work out the value of one share, then multiply back up.

Worked example Share $360 between Aisha, Ben and Carla in the ratio 2 : 3 : 4.
Total parts: 2 + 3 + 4 = 9
Value of one part: 360 ÷ 9 = 40
Aisha: 2 × 40 = $80   Ben: 3 × 40 = $120   Carla: 4 × 40 = $160
(Check: 80 + 120 + 160 = 360 ✓)
Try it — Dividing in a Ratio
  1. Divide 45 kg in the ratio 4 : 5.
  2. Divide $200 in the ratio 3 : 2.
  3. Divide 96 sweets between three children in the ratio 1 : 3 : 4.
  4. A concrete mix uses cement, sand and gravel in the ratio 1 : 2 : 3. How much cement is needed to make 60 kg of the mix?
Show answers
  1. 20 kg and 25 kg
  2. $120 and $80
  3. 12, 36, 48 sweets
  4. Total parts = 6, one part = 10 kg, so 10 kg of cement

3. Ratio in Context — Scales, Recipes and Best Value

Worked example — map scales A map has a scale of 1 : 25,000. Two towns are 4 cm apart on the map. What is the actual distance?
4 × 25,000 = 100,000 cm = 1000 m = 1 km
Worked example — best value Shop A sells 500 g of coffee for $3.50. Shop B sells 750 g for $5.10. Which is better value?
Price per gram, Shop A: 3.50 ÷ 500 = $0.007/g
Price per gram, Shop B: 5.10 ÷ 750 ≈ $0.0068/g
Shop B is cheaper per gram, so Shop B offers better value.
Try it — Ratio in Context
  1. A map has a scale of 1 : 50,000. Two points are 6 cm apart on the map. What is the actual distance, in km?
  2. A recipe for 4 people uses 250 g of flour. How much flour is needed for 6 people?
  3. Shop X sells 6 eggs for $1.80. Shop Y sells 10 eggs for $2.90. Which is better value per egg?
Show answers
  1. 6 × 50,000 = 300,000 cm = 3 km
  2. 250 × 6/4 = 375 g
  3. Shop X: $0.30/egg. Shop Y: $0.29/egg. Shop Y is better value.

4. Proportional Reasoning — The Unitary Method

The unitary method means finding the value of a single unit first, then scaling to whatever amount you actually need. It works for direct proportion — when quantities increase or decrease together at the same rate.

Worked example 5 pens cost $12. What is the cost of 8 pens?
Cost of 1 pen: 12 ÷ 5 = $2.40
Cost of 8 pens: 8 × 2.40 = $19.20
Worked example 9 identical books weigh 3.6 kg in total. What would 15 of the same books weigh?
Weight of 1 book: 3.6 ÷ 9 = 0.4 kg
Weight of 15 books: 15 × 0.4 = 6 kg
Try it — Unitary Method
  1. 7 identical chairs cost $210. What is the cost of 12 chairs?
  2. 4 workers can build a wall in 10 days. Using the unitary method idea, if a job needs 40 worker-days of effort, how many days would 8 workers take? (Hint: think about total work first.)
  3. A car uses 6 litres of fuel to travel 90 km. How far can it travel on 15 litres, at the same rate?
Show answers
  1. Cost of 1 chair = 210 ÷ 7 = $30, so 12 chairs = $360
  2. 4 workers × 10 days = 40 worker-days total. 40 worker-days ÷ 8 workers = 5 days
  3. 90 ÷ 6 = 15 km per litre; 15 × 15 = 225 km

5. Common Measures of Rate

A rate compares two quantities measured in different units — that’s what makes it different from a ratio. Pay rates, exchange rates, flow rates and fuel consumption are all rates you’ll meet regularly.

Worked example — hourly pay A tutor is paid $15 per hour. How much do they earn for 7 hours of work?
7 × 15 = $105
Worked example — currency exchange The exchange rate is 1 USD = 83 INR. Convert 250 USD to INR, and separately convert 4150 INR to USD.
250 × 83 = 20,750 INR
4150 ÷ 83 = 50 USD
Worked example — flow rate and fuel consumption A tap fills a tank at 3 litres per minute. How long does it take to fill a 90-litre tank?
90 ÷ 3 = 30 minutes

A car travels 15 km per litre of fuel. How many litres are needed for a 300 km journey?
300 ÷ 15 = 20 litres
Try it — Rates
  1. A worker is paid $12.50 per hour. How much do they earn for a 6-hour shift?
  2. The exchange rate is 1 GBP = 105 INR. Convert 40 GBP to INR.
  3. A hose fills a paddling pool at 8 litres per minute. How long to fill a 200-litre pool?
  4. A motorbike travels 35 km per litre. How far can it travel on 12 litres?
Show answers
  1. $75
  2. 4,200 INR
  3. 25 minutes
  4. 420 km

6. Density, Pressure and Population Density

These rates connect two different quantities through a formula. Good news: for these three, the formula will be given to you in the question — you don’t need to memorise them, just apply them correctly.

Density = Mass ÷ Volume
Population density = Population ÷ Area
Worked example — density A metal block has a mass of 270 g and a volume of 30 cm³. Find its density.
Density = 270 ÷ 30 = 9 g/cm³
Worked example — population density A region has a population of 1,200,000 and an area of 400 km². Find the population density.
1,200,000 ÷ 400 = 3,000 people per km²
Try it — Density & Population Density
  1. A block has mass 500 g and volume 40 cm³. Find its density.
  2. A liquid has a density of 1.2 g/cm³ and a volume of 50 cm³. Find its mass. (Rearrange the formula first.)
  3. A city has a population of 850,000 and covers 250 km². Find the population density.
Show answers
  1. 500 ÷ 40 = 12.5 g/cm³
  2. Mass = Density × Volume = 1.2 × 50 = 60 g
  3. 850,000 ÷ 250 = 3,400 people per km²

7. Speed, Distance and Time

Unlike density and population density, the speed/distance/time relationship is not given in the exam — you’re expected to know it.

Speed = Distance ÷ Time

Rearranged: Distance = Speed × Time, and Time = Distance ÷ Speed. A useful memory tool: arrange D, S, T in a triangle with D on top and S, T on the bottom — cover the one you want to find, and the triangle shows you the other two combined the right way.

Worked example A car travels 180 km in 2.5 hours. Find its average speed.
Speed = 180 ÷ 2.5 = 72 km/h
Worked example — converting mixed time A cyclist travels 54 km in 2 hours 15 minutes. Find their average speed in km/h.
Convert the time to hours: 2 hours 15 minutes = 2.25 hours
Speed = 54 ÷ 2.25 = 24 km/h
Worked example — average speed over multiple stages A runner covers 60 km in 1.5 hours, then a further 40 km in 1 hour. Find their average speed for the whole journey.
Total distance: 60 + 40 = 100 km. Total time: 1.5 + 1 = 2.5 hours.
Average speed = 100 ÷ 2.5 = 40 km/h
Note: this is not the same as averaging the two individual speeds (40 km/h and 40 km/h happen to match here, but that won’t usually be the case) — always use total distance ÷ total time.
Worked example — converting units of speed Convert 90 km/h to m/s.
90 km/h = 90,000 m / 3600 s = 25 m/s

Convert 18 m/s to km/h.
18 m/s × 3600 ÷ 1000 = 64.8 km/h
Try it — Speed, Distance, Time
  1. A train travels 240 km in 3 hours. Find its average speed.
  2. A cyclist travels 30 km at an average speed of 12 km/h. How long does the journey take?
  3. A car travels at 80 km/h for 3.5 hours. How far does it travel?
  4. A walker covers 21 km in 3 hours 30 minutes. Find their average speed in km/h.
  5. Convert 20 m/s to km/h.
  6. A journey has two stages: 45 km in 1 hour, then 30 km in 1.5 hours. Find the average speed for the whole journey.
Show answers
  1. 80 km/h
  2. 30 ÷ 12 = 2.5 hours
  3. 80 × 3.5 = 280 km
  4. 3 hours 30 minutes = 3.5 hours; 21 ÷ 3.5 = 6 km/h
  5. 20 × 3600 ÷ 1000 = 72 km/h
  6. Total distance = 75 km, total time = 2.5 hours, average speed = 30 km/h
Common mistakes to watch for
  • Simplifying a ratio using the wrong factor, or forgetting to convert to the same units first (e.g. mixing cm and m before simplifying).
  • Dividing a quantity in a ratio but forgetting to add the parts first — you must know the total number of shares before finding the value of one share.
  • Averaging two speeds directly (adding them and dividing by 2) instead of using total distance ÷ total time — these only give the same answer by coincidence.
  • Forgetting to convert time into hours (or minutes) fully before dividing — 2 hours 15 minutes is 2.25 hours, not 2.15.
  • Mixing up which quantity is “per” which in a rate — always check the units given in the question (km/h vs h/km are very different things).

8. Quick-Fire Challenge Round

True or False?
  1. A ratio and a rate are the same thing.
  2. To divide a quantity in a ratio, you always divide by the number of terms in the ratio (e.g. always divide by 2 for a two-part ratio).
  3. The density formula will be given to you in the exam.
  4. The speed/distance/time formula will be given to you in the exam.
  5. Average speed for a whole journey equals the mean of the speeds of each stage.
Show answers
  1. False — a ratio compares quantities of the same kind (no units, or matching units); a rate compares quantities of different kinds (e.g. km per hour).
  2. False — you divide by the total number of parts, which depends on the ratio itself, not just how many terms it has (e.g. 2:3 has 5 parts, not 2).
  3. True.
  4. False — this one you’re expected to know.
  5. False — you must use total distance ÷ total time, not the average of the individual speeds.
Spot the Error
  1. “To share $100 in the ratio 1:4, each person gets $100 ÷ 2 = $50.”
  2. “A car travels 150 km in 2 hours, so its speed is 2 ÷ 150 = 0.013 km/h.”
  3. “90 km/h converted to m/s is 90 ÷ 3.6… wait, 90 × 3.6 = 324 m/s.”
Show answers
  1. The ratio 1:4 has 5 parts total, not 2 — one part = $100 ÷ 5 = $20, so the shares are $20 and $80.
  2. Speed = distance ÷ time, not time ÷ distance: 150 ÷ 2 = 75 km/h.
  3. To convert km/h to m/s, divide by 3.6, don’t multiply: 90 ÷ 3.6 = 25 m/s.

9. Mixed Practice — Bringing It All Together

  1. Simplify the ratio 36 : 48 : 60.
  2. Share $540 in the ratio 4 : 5.
  3. A map has a scale of 1 : 100,000. Two villages are 3.5 cm apart on the map. Find the real distance in km.
  4. A worker is paid $18.50 per hour and works 5.5 hours. Find their total pay.
  5. A block has mass 640 g and volume 80 cm³. Find its density.
  6. A cyclist rides 63 km in 3 hours 30 minutes. Find their average speed in km/h.
Show answers
  1. 3 : 4 : 5
  2. Total parts = 9, one part = $60, so $240 and $300
  3. 3.5 × 100,000 = 350,000 cm = 3.5 km
  4. 18.50 × 5.5 = $101.75
  5. 640 ÷ 80 = 8 g/cm³
  6. 3.5 hours; 63 ÷ 3.5 = 18 km/h

Practice at Home

A large set again — work through it in stages.

Part A — Simplifying ratios
  1. Simplify 24 : 36.
  2. Simplify 28 : 42 : 70.
  3. Write 600 m : 1.5 km as a ratio in simplest form.
  4. Write 20 minutes : 1 hour as a ratio in simplest form.
Part B — Dividing in a ratio
  1. Divide 72 kg in the ratio 5 : 3.
  2. Divide $450 in the ratio 2 : 3 : 4.
  3. A fruit drink mixes orange and mango juice in the ratio 3 : 2. How much orange juice is needed to make 2 litres of drink?
Part C — Ratio in context and unitary method
  1. A map has a scale of 1 : 20,000. Two points are 8 cm apart on the map. Find the actual distance in metres.
  2. A recipe for 5 people needs 400 g of rice. How much rice is needed for 8 people?
  3. 8 identical calculators cost $96. What would 13 of the same calculators cost?
  4. Shop P sells 12 eggs for $3.24. Shop Q sells 18 eggs for $4.68. Which is better value per egg?
Part D — Rates
  1. A worker earns $16.20 per hour. Find their pay for an 8-hour shift.
  2. The exchange rate is 1 EUR = 90 INR. Convert 65 EUR to INR, and separately convert 3600 INR to EUR.
  3. A pipe fills a tank at 12 litres per minute. How long to fill a 540-litre tank?
  4. A van travels 9 km per litre of fuel. How many litres are needed for a 216 km trip?
Part E — Density and population density
  1. A block has mass 918 g and volume 102 cm³. Find its density.
  2. A liquid has density 0.8 g/cm³. Find the mass of 250 cm³ of the liquid.
  3. A town has population 96,000 and area 32 km². Find its population density.
Part F — Speed, distance and time
  1. A car travels 315 km in 4.5 hours. Find its average speed.
  2. A plane flies at 720 km/h for 3 hours 15 minutes. Find the distance travelled.
  3. A hiker walks 18 km at an average speed of 4 km/h. How long does it take, in hours and minutes?
  4. Convert 108 km/h to m/s.
  5. Convert 15 m/s to km/h.
Part G — Applied / challenge
  1. A journey has three stages: 20 km in 30 minutes, 45 km in 45 minutes, and 15 km in 15 minutes. Find the average speed for the entire journey, in km/h.
  2. Two friends share a $720 lottery win in the ratio of the number of tickets they each bought: Friend A bought 5 tickets, Friend B bought 7 tickets. How much does each friend receive?
  3. A car’s fuel tank holds 45 litres. The car travels 12 km per litre. If a journey is 480 km long, is a full tank enough? If not, how many more litres are needed?

Key Vocabulary Recap

Ratio — compares quantities of the same kind, written a:b
Rate — compares quantities of different kinds, e.g. km per hour
Unitary method — finding the value of one unit first, then scaling up or down
Density — mass ÷ volume
Population density — population ÷ area
Average speed — total distance ÷ total time

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) sections C1.11 and C1.12 (Core subject content) and E1.11 and E1.12 (Extended subject content) — Ratio and proportion, and Rates. Unlike most Number topics, the wording of these sections is identical for both tiers, so no content here is tier-restricted.

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