Ratio and Proportion

Overview

Ratio and Proportion forms the backbone of quantitative aptitude in IBPS PO Prelims. This topic rarely appears as standalone questions but underpins nearly 40% of the paper—from partnership problems and mixture questions to data interpretation sets. Mastering ratios gives you speed advantages across multiple question types.

In IBPS PO Prelims, expect 2–4 direct questions on ratio-proportion, plus indirect applications in DI, profit-sharing, and age problems. The exam tests your ability to manipulate ratios quickly, convert between ratios and actual values, and apply compound ratios in multi-step problems. Speed and accuracy here directly impact your overall Quant score.

Your goal: develop intuition for ratio manipulation so that you spend minimal time on calculations and maximum time on complex puzzles and DI sets.

Key Concepts

• **Ratio basics**: A ratio a:b means for every 'a' parts of one quantity, there are 'b' parts of another. Always express in lowest terms by dividing by HCF.

• **Proportion**: When two ratios are equal (a:b = c:d), they form a proportion. Cross-multiplication gives ad = bc.

• **Componendo-Dividendo**: If a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d). Useful for quick solving when sums and differences are given.

• **Compound ratio**: When combining multiple ratios, multiply corresponding terms. Compound of a:b and c:d is ac:bd.

• **Duplicate and triplicate ratios**: Duplicate of a:b is a²:b². Triplicate is a³:b³. Sub-duplicate is √a:√b.

• **Distribution in a ratio**: If total T is divided in ratio a:b:c, first part = T × a/(a+b+c).

• **Ratio of increments**: If A:B = 3:4 and both increase by same amount x, new ratio changes. Set up equation with (A+x):(B+x) = new ratio.

• **Mean proportional**: Between a and b is √(ab). Third proportional to a,b is b²/a.

Formulas / Key Facts

| Concept | Formula/Rule | |---------|--------------| | Ratio to fraction | a:b means first quantity = a/(a+b) of total | | Cross multiplication | If a:b = c:d, then a×d = b×c | | Compound ratio | (a:b) × (c:d) × (e:f) = ace : bdf | | Duplicate ratio | Of a:b is a² : b² | | Sub-duplicate ratio | Of a:b is √a : √b | | Mean proportional | Of a and b = √(ab) | | Third proportional | To a and b = b²/a | | Componendo-Dividendo | (a+b)/(a−b) = (c+d)/(c−d) when a/b = c/d | | Partnership share | Profit share = Capital × Time invested | | Combining ratios | If A:B = 2:3 and B:C = 4:5, make B common → A:B:C = 8:12:15 |

Worked Examples

**Example 1: Basic Distribution**

₹2400 is divided among A, B, C in ratio 3:4:5. Find B's share.

*Solution:*

• Sum of ratio parts = 3 + 4 + 5 = 12
• B's share = (4/12) × 2400 = ₹800

**Example 2: Combining Ratios**

A:B = 2:3 and B:C = 5:7. Find A:B:C.

*Solution:*

• Make B common in both ratios
• In first ratio B = 3, in second B = 5
• LCM of 3 and 5 = 15
• Multiply first ratio by 5 → A:B = 10:15
• Multiply second ratio by 3 → B:C = 15:21
• Combined A:B:C = 10:15:21

**Example 3: Partnership Problem**

A invests ₹40,000 for 12 months. B invests ₹60,000 for 8 months. If total profit is ₹26,000, find A's share.

*Solution:*

• A's capital-time = 40,000 × 12 = 4,80,000
• B's capital-time = 60,000 × 8 = 4,80,000
• Ratio of profit = 4,80,000 : 4,80,000 = 1:1
• A's share = 26,000 × (1/2) = ₹13,000

**Example 4: Age-based Ratio**

Present ages of P and Q are in ratio 5:7. After 6 years, ratio becomes 3:4. Find present age of P.

*Solution:*

• Let present ages be 5x and 7x
• After 6 years: (5x + 6)/(7x + 6) = 3/4
• Cross multiply: 4(5x + 6) = 3(7x + 6)
• 20x + 24 = 21x + 18
• x = 6
• P's present age = 5 × 6 = 30 years

Common Mistakes

• **Adding ratios directly** → Wrong: A:B = 2:3, B:C = 3:4 doesn't give A:B:C = 2:3:4. Correct fix: Make the common term equal using LCM before combining.

• **Forgetting ratio is not actual value** → If ratio is 3:5, students assume actual values are 3 and 5. Correct fix: Actual values are 3k and 5k where k is determined from given total or difference.

• **Ignoring time in partnership** → Profit sharing depends on capital × time, not just capital. Correct fix: Always calculate capital-months before finding ratio.

• **Wrong cross-multiplication setup** → In proportions, students multiply wrong pairs. Correct fix: In a:b = c:d, product of extremes (a×d) = product of means (b×c).

• **Not simplifying before calculation** → Working with large numbers wastes time. Correct fix: Always reduce to lowest terms first—divide both terms by their HCF.

Quick Reference

• **Distribution formula**: Part = (its ratio / sum of ratios) × Total

• **Combining ratios**: Make common term equal using LCM

• **Partnership**: Profit ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

• **Age problems**: Let ages be ax and bx; use future/past condition to find x

• **Componendo-Dividendo**: When you know sum and difference, use (a+b)/(a−b)

• **Mean proportional of a and b = √(ab); Third proportional to a,b = b²/a**