Mixture and Alligation

Overview

Mixture and Alligation is a high-scoring topic in IBPS PO Prelims that appears both as standalone questions and within Data Interpretation sets. The concept helps you solve problems involving combinations of two or more ingredients at different prices, concentrations, or ratios—without setting up lengthy algebraic equations.

This topic connects directly with Ratio and Proportion, Percentage, and Profit-Loss. Mastering the alligation rule gives you a 30-second shortcut for problems that would otherwise take 2-3 minutes. Expect 1-2 direct questions in Prelims, often disguised as profit/loss or average-based word problems.

Students must be comfortable with: (1) the alligation cross method, (2) repeated replacement formula, and (3) converting between ratio and percentage forms quickly.

Key Concepts

• **Mixture**: A combination of two or more ingredients mixed together. Each ingredient has a characteristic value (price, concentration, percentage, etc.).

• **Alligation Rule**: When two ingredients at different values are mixed, the ratio of their quantities is inversely proportional to the difference of their values from the mean value.

• **Mean Price/Value**: The weighted average characteristic of the final mixture. It always lies between the values of the two ingredients.

• **Cheaper and Dearer**: In alligation problems, the ingredient with lower value is "cheaper" and the one with higher value is "dearer." The mean always falls between these two.

• **Replacement Problems**: When part of a mixture is removed and replaced with a pure ingredient, the concentration of the other ingredient decreases progressively with each operation.

• **Three-Component Mixtures**: Solve by applying alligation twice—first between two components, then combining with the third using the intermediate result.

Formulas / Key Facts

**The Alligation Rule (Cross Method)** ``` Quantity of Cheaper Dearer Value − Mean Value ─────────────────── = ───────────────────────── Quantity of Dearer Mean Value − Cheaper Value ``` Ratio = (d − m) : (m − c), where c = cheaper value, d = dearer value, m = mean value

**Repeated Replacement Formula** When a container has V litres of pure liquid A, and x litres is removed and replaced with liquid B, repeated n times:

Final quantity of A = V × (1 − x/V)ⁿ

Concentration of A after n operations = (1 − x/V)ⁿ × 100%

**Mixing Two Mixtures** If mixture 1 has components in ratio a:b and mixture 2 has components in ratio c:d, and they are mixed in ratio M:N, find the component-wise weighted average.