Ratio and Proportion

Introduction

Ratio and proportion form the foundation of many mathematical concepts in ICSE Class 10. A ratio compares two quantities of the same unit, while a proportion states that two ratios are equal. The properties of proportion — invertendo, alternendo, componendo, dividendo, and their combination — are particularly important for ICSE.

Basic Definitions

• Ratio — a : b = a / b, where a and b are quantities of the same kind.
• Proportion — a : b = c : d, written as a : b :: c : d.
• Extremes — In a : b :: c : d, a and d are extremes.
• Means — In a : b :: c : d, b and c are means.
• Product of extremes = Product of means: ad = bc.

Properties of Proportion

1. Invertendo

If a : b = c : d, then b : a = d : c.

a / b = c / d → b / a = d / c

2. Alternendo

If a : b = c : d, then a : c = b : d.

a / b = c / d → a / c = b / d

3. Componendo

If a : b = c : d, then (a + b) : b = (c + d) : d.

a / b = c / d → (a + b) / b = (c + d) / d

4. Dividendo

If a : b = c : d, then (a − b) : b = (c − d) : d.

a / b = c / d → (a − b) / b = (c − d) / d

5. Componendo and Dividendo (Combined)

If a : b = c : d, then (a + b) : (a − b) = (c + d) : (c − d).

a / b = c / d → (a + b) / (a − b) = (c + d) / (c − d)

This is the most powerful and frequently tested property in ICSE exams.

Continued Proportion

Three quantities a, b, c are said to be in continued proportion if:

a : b = b : c

i.e., a / b = b / c → b² = ac

Here, b is called the mean proportional between a and c, and c is called the third proportional to a and b.

Worked Examples

Example 1: Using Componendo and Dividendo

If (3x + 5y) / (3x − 5y) = 7 / 3, find x : y.

Solution: Using componendo and dividendo: (3x + 5y + 3x − 5y) / (3x + 5y − 3x + 5y) = (7 + 3) / (7 − 3) (6x) / (10y) = 10 / 4 6x / 10y = 5 / 2 3x / 5y = 5 / 2 Cross multiply: 6x = 25y x / y = 25 / 6 x : y = 25 : 6

Example 2: Finding Mean Proportional

Find the mean proportional between 16 and 25.

Solution: Let the mean proportional be b. 16 / b = b / 25 b² = 16 × 25 = 400 b = √400 = ±20

Since mean proportional is usually taken as positive, b = 20.

Example 3: Third Proportional

Find the third proportional to 12 and 18.

Solution: Let the third proportional be c. 12 : 18 = 18 : c 12 / 18 = 18 / c 12c = 324 c = 324 / 12 = 27

Example 4: Complex Application

If a : b = 3 : 4 and b : c = 5 : 6, find a : b : c.

Solution: a : b = 3 : 4 = 15 : 20 (multiply by 5) b : c = 5 : 6 = 20 : 24 (multiply by 4)

Therefore, a : b : c = 15 : 20 : 24

Application: Verifying Properties

Given a : b = 3 : 5 and b : c = 5 : 7, verify that a : c = 3 : 7 (Alternendo property).

Solution: a : b = 3 : 5 → a / b = 3 / 5 b : c = 5 : 7 → b / c = 5 / 7

Multiplying: (a / b) × (b / c) = (3 / 5) × (5 / 7) a / c = 3 / 7 Hence a : c = 3 : 7. ✓ Verified.

Common Mistakes and Fixes

ICSE Exam Focus

Ratio and proportion carry 6–10 marks in ICSE examinations. Key question types:

• Applying componendo and dividendo to simplify expressions.
• Finding mean proportional, third proportional.
• Comparing ratios.
• Word problems on ratios and proportions.
• Combined ratio (a : b : c).

Marks Blueprint:

Self-Test Questions

1. If (7x − 3y) / (7x + 3y) = 3 / 5, find x : y using componendo and dividendo.

2. Find the mean proportional between 8 and 32.

3. Find the third proportional to 9 and 15.

4. If a : b = 2 : 3, b : c = 4 : 5, and c : d = 6 : 7, find a : b : c : d.

5. If (2a + 3b) / (3a − 2b) = 5 / 2, find a : b.

6. State and prove the componendo and dividendo property.

The combined componendo-dividendo property is a favourite in ICSE exams — mastering it will save you time and earn full marks on proportion problems.