Module 2 · IBPS PO Quantitative Aptitude

Ratio, Proportion & Mixture

Ratio, mixture & alligation, partnership, coin problems.

Ratio · Mixture & Alligation · Partnership

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Learning Objectives

Understand the fundamental concepts of Ratio, Proportion & Mixture
Apply key formulas and techniques to solve problems
Practice with exam-level questions to build speed and accuracy

Key Concepts

What is Ratio, Proportion & Mixture?

Ratio compares two quantities of the same unit. Proportion states that two ratios are equal. Mixture and alligation problems deal with combining two or more ingredients at different prices or concentrations. These topics are frequently tested in SSC CGL Tier 1.

Key Concepts

1. Ratio: a:b means a/b. If a:b = c:d, then ad = bc (cross-multiplication). A ratio is in its simplest form when a and b have no common factors.

2. Proportion: If a:b = c:d, then a, b, c, d are in proportion. The product of extremes = product of means (a*d = b*c).

3. Componendo & Dividendo: If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d). This is useful for solving equations involving ratios.

4. Alligation Rule: Used to find the ratio in which two ingredients at different prices are mixed. (Cheaper quantity) : (Dearer quantity) = (Mean price - Cheaper) : (Dearer - Mean price).

5. Mixture Replacement: When a quantity is removed from a mixture and replaced with another, the remaining fraction of the original ingredient = 1 - (replaced quantity / total quantity).

Solved Examples

Example 1 (Basic Ratio)

If A:B = 2:3 and B:C = 5:7, find A:C.

a) 10:21 b) 10:7 c) 2:7 d) 3:7

Solution: a) 10:21. Make B common: A:B = 2:3 = 10:15, B:C = 5:7 = 15:21. So A:C = 10:21.

Example 2 (Alligation)

In what ratio must rice at Rs.60/kg be mixed with rice at Rs.90/kg to get a mixture worth Rs.75/kg?

a) 1:1 b) 2:1 c) 1:2 d) 3:2

Solution: a) 1:1. Using alligation: (90-75):(75-60) = 15:15 = 1:1.

Example 3 (Mixture Replacement)

From a 20L vessel full of milk, 4L is removed and replaced with water. This is repeated once more. Find the milk remaining.

a) 10.8 L b) 12.8 L c) 14.4 L d) 16 L

Solution: b) 12.8 L. After each operation, remaining fraction = 1 - 4/20 = 4/5. After 2 operations, milk = 20 * (4/5)^2 = 20 * 16/25 = 12.8 L.

Shortcut Techniques

To find A:C from A:B and B:C, make B equal in both ratios by finding LCM of B values.
Alligation is the fastest method for mixture questions. Draw a simple cross diagram.
For successive replacements, use the formula: Final quantity = Initial * (1 - r/n)^k, where r is replaced quantity, n is total, k is number of operations.
For income/expenditure ratio problems, use the common difference method to find individual values.

Pro Tip: The alligation cross method is extremely versatile — use it not just for prices but also for percentages, speeds, concentrations, and any weighted average problem.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Practice Questions

← Simple & Compound Interest Mensuration →