IGCSE Ratios and Proportions: Complete Guide for Cambridge IGCSE Mathematics

IGCSE ratios and proportions are essential topics in Cambridge IGCSE Mathematics that appear frequently in both Paper 2 and Paper 4. Mastering ratio simplification, direct proportion, and inverse proportion is crucial for solving real-world problems involving comparisons, scaling, and relationships between quantities.

This comprehensive IGCSE ratios and proportions guide covers everything you need to know, including writing and simplifying ratios, sharing quantities in ratios, direct and inverse proportion problems, step-by-step worked examples, common exam questions, and expert tips from Tutopiya’s IGCSE maths tutors. We’ll also show you how to avoid the most common mistakes that cost students valuable marks.

🎯 What you’ll learn: By the end of this guide, you’ll know how to simplify ratios, share quantities in given ratios, solve direct and inverse proportion problems, and apply these skills to real-world scenarios in IGCSE exams.

Already studying with Tutopiya? Practice these skills with our dedicated IGCSE Ratio and Proportion practice deck featuring exam-style questions and instant feedback.

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Why IGCSE Ratios and Proportions Matter

IGCSE ratios and proportions are fundamental concepts with wide applications. Here’s why they’re so important:

• High frequency topic: Ratio and proportion questions appear in almost every IGCSE maths paper
• Real-world applications: Used in cooking, maps, scale drawings, currency exchange, and many practical scenarios
• Exam weight: Typically worth 6-10 marks per paper
• Foundation for advanced topics: Essential for understanding similarity, trigonometry, and rates of change
• Problem-solving skills: Develops proportional reasoning and algebraic manipulation

Key insight from examiners: Students often confuse direct and inverse proportion, or make errors when simplifying ratios. This guide will help you master these concepts systematically.

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Understanding Ratios: The Basics

A ratio compares two or more quantities, showing how many times one quantity is contained in another.

Notation:

• Ratios are written using a colon : or as a fraction
• Example: 3:5 or 3/5 means “3 to 5”

Key points:

• Ratios don’t have units (they’re just numbers)
• Ratios should be simplified to their lowest terms
• The order matters: 3:5 is different from 5:3

Writing Ratios

Example: In a class of 30 students, 18 are girls and 12 are boys.

• Ratio of girls to boys: 18:12 or simplified 3:2
• Ratio of boys to girls: 12:18 or simplified 2:3
• Ratio of girls to total: 18:30 or simplified 3:5

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Simplifying Ratios

Method 1: Divide by the Highest Common Factor (HCF)

Example: Simplify 24:36

Solution:

1. Find HCF of 24 and 36: HCF = 12
2. Divide both parts by 12: 24 ÷ 12 : 36 ÷ 12 = 2:3

Answer: 2:3

Method 2: Divide by Common Factors Step by Step

Example: Simplify 48:72

Solution:

1. Both divisible by 8: 48 ÷ 8 : 72 ÷ 8 = 6:9
2. Both divisible by 3: 6 ÷ 3 : 9 ÷ 3 = 2:3

Answer: 2:3

Ratios with Decimals

Method: Multiply both parts by the same power of 10 to eliminate decimals, then simplify.

Example: Simplify 1.5:2.5

Solution:

1. Multiply by 10: 15:25
2. Simplify: 15:25 = 3:5 (divide by 5)

Answer: 3:5

Ratios with Fractions

Method: Multiply both parts by the LCM of the denominators to eliminate fractions, then simplify.

Example: Simplify 1/2 : 3/4

Solution:

1. LCM of 2 and 4 is 4
2. Multiply by 4: (1/2 × 4) : (3/4 × 4) = 2:3

Answer: 2:3

Ratios with Mixed Numbers

Example: Simplify 2 1/2 : 1 1/4

Solution:

1. Convert to improper fractions: 5/2 : 5/4
2. Multiply by LCM of denominators (4): (5/2 × 4) : (5/4 × 4) = 10:5
3. Simplify: 10:5 = 2:1

Answer: 2:1

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Sharing Quantities in a Given Ratio

When sharing a quantity in a given ratio, you need to divide it into parts proportional to the ratio.

Method

1. Add the ratio parts to find the total number of parts
2. Divide the quantity by the total number of parts to find the value of one part
3. Multiply each part of the ratio by the value of one part

Example: Share $120 in the ratio 3:5

Solution:

1. Total parts: 3 + 5 = 8
2. Value of one part: $120 ÷ 8 = $15
3. First share: 3 × $15 = $45
4. Second share: 5 × $15 = $75

Check: $45 + $75 = $120 ✓

Answer: $45 and $75

Example: Three-Part Ratio

Share 240 sweets in the ratio 2:3:7

Solution:

1. Total parts: 2 + 3 + 7 = 12
2. Value of one part: 240 ÷ 12 = 20
3. First share: 2 × 20 = 40
4. Second share: 3 × 20 = 60
5. Third share: 7 × 20 = 140

Check: 40 + 60 + 140 = 240 ✓

Answer: 40, 60, and 140 sweets

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Finding Missing Values in Ratios

Sometimes you’re given part of a ratio and need to find the other part(s).

Example: If the ratio of apples to oranges is 3:5 and there are 15 apples, how many oranges are there?

Solution:

1. Apples represent 3 parts, and there are 15 apples
2. Value of one part: 15 ÷ 3 = 5
3. Oranges represent 5 parts: 5 × 5 = 25

Answer: 25 oranges

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Understanding Proportions

Proportion describes the relationship between two ratios that are equal.

Notation: a:b = c:d or a/b = c/d

This means the ratios are equivalent.

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Direct Proportion

In direct proportion, as one quantity increases, the other increases by the same factor (and vice versa).

Key relationship: y ∝ x means y = kx where k is a constant

Method for Solving Direct Proportion Problems

1. Set up the proportion: y = kx or y₁/x₁ = y₂/x₂
2. Find the constant k if needed
3. Use the relationship to find the unknown

Example 1: If 5 apples cost $3, how much do 8 apples cost?

Solution:

1. Set up proportion: Cost/Apples = constant
2. $3/5 = Cost/8
3. Cross multiply: Cost = ($3 × 8)/5 = $24/5 = $4.80

Answer: $4.80

Alternative method:

1. Cost per apple: $3 ÷ 5 = $0.60
2. Cost of 8 apples: 8 × $0.60 = $4.80

Example 2: y is directly proportional to x. When x = 4, y = 12. Find y when x = 7.

Solution:

1. y = kx
2. Find k: 12 = k × 4, so k = 3
3. When x = 7: y = 3 × 7 = 21

Answer: y = 21

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Inverse Proportion

In inverse proportion, as one quantity increases, the other decreases by the same factor (and vice versa).

Key relationship: y ∝ 1/x means y = k/x where k is a constant

Method for Solving Inverse Proportion Problems

1. Set up the relationship: y = k/x or y₁x₁ = y₂x₂
2. Find the constant k if needed
3. Use the relationship to find the unknown

Example 1: If 6 workers can complete a job in 8 days, how long will it take 4 workers?

Solution:

1. This is inverse proportion (more workers = fewer days)
2. Set up: Workers × Days = constant
3. 6 × 8 = 4 × Days
4. 48 = 4 × Days
5. Days = 48 ÷ 4 = 12

Answer: 12 days

Example 2: y is inversely proportional to x. When x = 3, y = 8. Find y when x = 6.

Solution:

1. y = k/x
2. Find k: 8 = k/3, so k = 24
3. When x = 6: y = 24/6 = 4

Answer: y = 4

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Step-by-Step Method for Proportion Problems

1. Identify the type - Direct or inverse proportion?
2. Set up the relationship:
   • Direct: y = kx or y₁/x₁ = y₂/x₂
   • Inverse: y = k/x or y₁x₁ = y₂x₂
3. Find the constant k if needed
4. Solve for the unknown
5. Check your answer - Does it make sense?

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Worked Examples

Example 1: Simplifying Ratios

Simplify 0.8:1.2

Solution:

1. Multiply by 10: 8:12
2. Simplify: 8:12 = 2:3 (divide by 4)

Answer: 2:3

Example 2: Sharing in a Ratio

A sum of money is shared between Alice, Bob, and Charlie in the ratio 4:5:6. If Bob receives $150, find the total amount shared.

Solution:

1. Bob’s share represents 5 parts = $150
2. Value of one part: $150 ÷ 5 = $30
3. Total parts: 4 + 5 + 6 = 15
4. Total amount: 15 × $30 = $450

Answer: $450

Example 3: Direct Proportion

The distance traveled is directly proportional to time. A car travels 240 km in 3 hours. How far will it travel in 5 hours?

Solution:

1. Distance/Time = constant
2. 240/3 = Distance/5
3. 80 = Distance/5
4. Distance = 80 × 5 = 400 km

Answer: 400 km

Example 4: Inverse Proportion

The time taken to complete a task is inversely proportional to the number of workers. If 5 workers take 12 days, how long will 6 workers take?

Solution:

1. Workers × Days = constant
2. 5 × 12 = 6 × Days
3. 60 = 6 × Days
4. Days = 10

Answer: 10 days

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Common Examiner Traps (and How to Dodge Them)

• Confusing direct and inverse proportion - Direct: both increase together. Inverse: one increases, the other decreases
• Not simplifying ratios - Always simplify to lowest terms
• Order of ratio matters - 3:5 is different from 5:3
• Forgetting to check - Always verify your answer makes sense
• Units in ratios - Ratios don’t have units, but the quantities being compared do
• Inverse proportion formula - Remember: y₁x₁ = y₂x₂ for inverse proportion

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IGCSE Ratios and Proportions Practice Questions

Question 1: Simplifying Ratios

Simplify: a) 16:24 b) 1.5:2.25 c) 2/3 : 5/6

Solution: a) 16:24 = 2:3 (divide by 8) b) Multiply by 100: 150:225 = 2:3 (divide by 75) c) Multiply by 6: 4:5

Answers: a) 2:3 b) 2:3 c) 4:5

Question 2: Sharing in a Ratio

$360 is shared between three people in the ratio 3:4:5. How much does each person receive?

Solution:

1. Total parts: 3 + 4 + 5 = 12
2. One part: $360 ÷ 12 = $30
3. First: 3 × $30 = $90
4. Second: 4 × $30 = $120
5. Third: 5 × $30 = $150

Answer: $90, $120, $150

Question 3: Direct Proportion

y is directly proportional to x². When x = 2, y = 12. Find y when x = 5.

Solution:

1. y = kx²
2. 12 = k × 2² = 4k, so k = 3
3. When x = 5: y = 3 × 5² = 3 × 25 = 75

Answer: y = 75

Question 4: Inverse Proportion

The pressure of a gas is inversely proportional to its volume. When the volume is 8 m³, the pressure is 6 Pa. Find the pressure when the volume is 12 m³.

Solution:

1. Pressure × Volume = constant
2. 6 × 8 = Pressure × 12
3. 48 = Pressure × 12
4. Pressure = 4 Pa

Answer: 4 Pa

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Frequently Asked Questions About IGCSE Ratios and Proportions

What is a ratio?

A ratio compares two or more quantities, showing how many times one quantity is contained in another. Example: 3:5 means 3 parts to 5 parts.

How do I simplify a ratio?

Divide both parts by their Highest Common Factor (HCF). Example: 24:36 = 2:3 (divide by 12).

What’s the difference between direct and inverse proportion?

• Direct proportion: As one quantity increases, the other increases (e.g., more apples = more cost)
• Inverse proportion: As one quantity increases, the other decreases (e.g., more workers = fewer days)

How do I share a quantity in a ratio?

1. Add the ratio parts to get total parts
2. Divide the quantity by total parts to get one part
3. Multiply each ratio part by the value of one part

What does y ∝ x mean?

y ∝ x means “y is directly proportional to x”, which means y = kx for some constant k.

What does y ∝ 1/x mean?

y ∝ 1/x means “y is inversely proportional to x”, which means y = k/x for some constant k.

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Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Fractions, Decimals and Percentages: Complete Guide - Master conversions and calculations
• IGCSE Directed Numbers: Complete Guide - Master positive and negative numbers
• IGCSE Maths Revision Notes, Syllabus and Preparation Tips - Complete syllabus overview, topic breakdown, and revision strategies
• IGCSE Past Papers Guide - Access free IGCSE past papers and exam resources

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