Ratio, Proportion & Rates
CIE IGCSE Mathematics· 1.11, 1.12· 25 min read
1. Simplifying Ratios & Dividing Quantities★★☆☆☆⏱ 7 min
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Ratio
A comparison of two or more quantities of the same unit, written as a:b:c for three or more terms.
Example:
The ratio of 6 apples to 9 oranges is 6:9, which simplifies to 2:3.
To simplify a ratio, divide all terms by their highest common factor (HCF). Always convert all terms to the same unit before simplifying, for example a ratio of 200g to 1kg first becomes 200g:1000g, simplified to 1:5. To split a quantity in a given ratio, first calculate the total number of parts, find the value of 1 part, then multiply by each ratio term.
Simplify the ratio 12:18:30, then split \$180 in this simplified ratio.
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Step 1: Find the HCF of 12, 18 and 30, which is 6.
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Step 2: Divide all terms by 6: 12/6 : 18/6 : 30/6 = 2:3:5
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Step 3: Calculate total number of parts: 2 + 3 + 5 = 10 parts
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Step 4: Find value of 1 part: 180 ÷ 10 = \$18
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Step 5: Multiply each ratio term by 1 part: 2×18 = \54, 5×18 = \$90
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Check: 36 + 54 + 90 = 180, matching the original total amount.
Exam tip:
Always confirm units are identical for all terms before simplifying ratios, as this is one of the most common mark-losing errors on Core ratio questions.
2. Proportional Reasoning for Real-World Problems★★☆☆☆⏱ 6 min
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Proportional reasoning uses equivalent ratios to find unknown quantities. The unitary method (finding the value of 1 unit first) is the most reliable approach for Core questions, even if you know other techniques.
5 identical notebooks cost \$3.75. Calculate the cost of 12 identical notebooks.
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Step 1: Find the cost of 1 notebook: 3.75 ÷ 5 = \$0.75
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Step 2: Multiply by 12 to get total cost: 0.75 × 12 = \$9.00
If 4 litres of paint cover 16 square metres of wall, how many litres do you need to cover 40 square metres?
Reveal answer
10 litres —1 litre covers 4 m², so 40 ÷ 4 = 10 litres required.
3. Calculating Common Rates★★★☆☆⏱ 6 min
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Rate
A comparison of two quantities with different units, often expressed as 'per unit' of one quantity.
Example:
Hourly pay is measured in dollars per hour (\$/h), fuel consumption in km per litre (km/l).
Common rates tested in Core exams include hourly pay, currency exchange rates, fuel consumption, and flow rate (litres per minute). Always check the units required in your final answer to avoid unnecessary conversion errors.
A car travels 315 km using 45 litres of petrol. Calculate its fuel consumption in km per litre.
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Step 1: Fuel consumption = total distance ÷ total petrol used
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Final answer: 7 km per litre.
4. Average Speed, Density & Pressure★★★☆☆⏱ 6 min
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These three rate measures are frequently tested in Core papers. The formulae for density and pressure are provided on your exam formula sheet, but memorising the average speed formula will save you time during the exam.
A cyclist travels 24 km in 1 hour 30 minutes. Calculate her average speed in km/h.
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Step 1: Convert time to hours only: 1 hour 30 minutes = 1.5 hours
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Step 2: Use average speed = total distance ÷ total time
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Final answer: 16 km/h
A metal block has a mass of 1800 g and a volume of 200 cm³. Use the formula density = mass / volume to calculate its density in g/cm³.
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Step 1: Substitute values into the given formula
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Final answer: 9 g/cm³
Exam tip:
Double-check that units of the given values match the units requested in the answer, especially for time (hours vs minutes) when calculating average speed.
5. Common Pitfalls
Wrong move:
Simplifying 500g : 2kg directly as 250:1 without converting units first.
Why:
You used different units for the ratio terms, leading to an inverted, incorrect result.
Correct move:
Convert 2kg to 2000g first, then simplify 500:2000 to 1:4.
Wrong move:
Splitting \40 and \$70.
Why:
You forgot to sum the ratio terms to get total parts, so you miscalculated the value of 1 part.
Correct move:
Sum ratio terms first (2+3=5), value of 1 part = 100/5 = \40 and \$60.
Wrong move:
Calculating average speed as the average of two individual speeds for a journey.
Why:
Average speed depends on total distance and total time, not the average of separate speeds.
Correct move:
Calculate total distance travelled, divide by total time taken for the entire journey.
Wrong move:
Calculating density as volume divided by mass.
Why:
You mixed up the numerator and denominator of the given density formula.
Correct move:
Use the given exam formula: density = mass / volume, rearrange only if needed for the unknown value.
Wrong move:
Using time in minutes to calculate average speed in km/h.
Why:
Units of time must match the denominator unit requested in the answer.
Correct move:
Convert minutes to hours by dividing by 60 before performing the calculation.
6. Quick Reference Cheatsheet
Concept | Formula / Method | Key Check |
|---|---|---|
Simplify ratio | Divide all terms by HCF | All terms have identical units first |
Split quantity in ratio | Total parts = sum of ratio terms; 1 part = total / total parts | Sum of final amounts matches original total |
Proportional reasoning | Unitary method: find value of 1 unit first | Units are consistent across calculation |
Average speed | Total distance / Total time | Time unit matches speed denominator (e.g. hours for km/h) |
Density | Mass / Volume (given in exam) | Units match requested output (e.g. g/cm³) |
Pressure | Force / Area (given in exam) | Area units are consistent across calculation |
7. Frequently Asked
Do I need to memorise the density and pressure formulae for Core?
No, these formulae are provided on the front of your exam paper. You only need to memorise the average speed formula for Core.
Can I simplify ratios with different units directly?
No, you must convert all terms to the same unit first before simplifying a ratio, otherwise your result will be incorrect.
Going deeper
What's Next
Now that you have mastered Core ratio, proportion and rate skills for CIE IGCSE 0580, you can move on to applying these skills to multi-step number problem-solving questions that combine multiple Core concepts. If you are studying for the Extended tier, you will next learn algebraic direct and inverse proportion in the Algebra unit, which builds on the proportional reasoning skills you have practised here. Make sure to complete past paper questions on this topic to reinforce your understanding and avoid common errors before your exam.