Mixture and Allegation — Study Notes

Overview

Mixture and Allegation is a staple arithmetic topic in SSC CHSL Tier 1 that typically yields 1–2 questions per exam. The core idea is combining two or more items (liquids, metals, grains, prices) in certain ratios and determining either the resultant mixture properties or the original proportions. The *allegation rule* is a shortcut method to find the ratio in which two ingredients must be mixed to achieve a desired mean value—avoiding lengthy algebraic equations.

This topic intersects with ratio-proportion and weighted average concepts. You must be comfortable with fraction manipulation and cross-multiplication. Most questions fall into three patterns: *mixing two components to get a mean price or concentration*, *repeated replacement of a mixture*, and *removing and adding mixtures*. Mastery here saves valuable seconds on exam day because allegation bypasses multi-step algebra. Expect problems on milk-water mixtures, alloy compositions and cost-price averaging of goods.

Understanding the allegation diagram—a visual cross-method—is essential. It lets you set up the ratio in one swift step. Replacement problems demand careful tracking of what remains versus what is added back. Practice is crucial because small arithmetic slips (like forgetting to square the replacement fraction) cost marks.

Key Concepts

• **Mixture**: A combination of two or more ingredients in definite or variable proportions. The *mean value* (price, concentration, etc.) lies between the individual ingredient values.
• **Allegation Rule**: A shortcut to find the ratio of two components when their individual values and the mixture's mean value are known. The ratio equals (Difference of higher value from mean) : (Difference of mean from lower value).
• **Alligation Diagram**: Draw a cross. Put the mean at the centre, the two ingredient values at the left corners, and compute the absolute differences diagonally. These differences give the required ratio.
• **Replacement Formula**: When a fraction *k* of a mixture is removed and replaced with a pure ingredient *n* times, the quantity of the original component after *n* operations is `Original × (1 − k)ⁿ`.
• **Weighted Average Principle**: If quantities Q₁ and Q₂ at values V₁ and V₂ mix, the mean value M = (Q₁V₁ + Q₂V₂) / (Q₁ + Q₂). Allegation bypasses this algebra when finding Q₁ : Q₂.
• **Two-Component Standard Setup**: Cheaper + Dearer → Mean price; Lower concentration + Higher concentration → Mean concentration; Pure substance + Mixture → Resultant concentration.
• **Ratio Inversion**: If the allegation gives ratio of quantities as a : b, sometimes the question asks "in what ratio to mix" and sometimes "what is the fraction of one component"—read carefully.
• **Unit Consistency**: Always check units (litres vs. ml, ₹ per kg vs. per gram). Convert before applying allegation.

Formulas / Key Facts

1. **Allegation Ratio Formula**: If ingredient A costs/concentration C₁ and ingredient B costs/concentration C₂ mix to give mean M, then Quantity A : Quantity B = (C₂ − M) : (M − C₁). *Remember*: higher value minus mean on the opposite side, lower value subtracted from mean on its opposite side.

2. **Replacement after n Draws**: Final quantity of original component = Initial quantity × (1 − k)ⁿ, where k = fraction replaced each time, n = number of times.

3. **Milk-Water Ratio Conversion**: If milk:water = a:b, then milk fraction = a/(a+b) and water fraction = b/(a+b). Use these fractions in replacement problems.

4. **Mean Price from Quantities**: Mean = (Q₁P₁ + Q₂P₂) / (Q₁ + Q₂). Allegation reverses this to find Q₁:Q₂ given P₁, P₂, Mean.

5. **Combined Ratio**: When mixing three components, apply allegation pairwise or use weighted average algebra carefully.

6. **Pure Substance Concentration**: Pure milk = 100% concentration; pure water = 0% concentration. Use these in concentration-based allegation.

7. **Inverse Proportionality**: Quantity mixed is inversely proportional to the price/concentration difference from the mean.

8. **Volume Conservation**: Total volume before = Total volume after, unless explicitly stated that volume changes on mixing.

Worked Examples

**Example 1 (Allegation — Mean Price)** *A shopkeeper mixes rice at ₹40/kg with rice at ₹60/kg to sell the mixture at ₹50/kg. In what ratio must he mix the two varieties?*

• Step 1: Identify C₁ = 40, C₂ = 60, M = 50.
• Step 2: Apply allegation rule.

Cheaper rice difference from mean = 50 − 40 = 10. Dearer rice difference from mean = 60 − 50 = 10.

• Step 3: Ratio = 10 : 10 = 1 : 1.
• Answer: He must mix them in **1:1 ratio**.

**Example 2 (Replacement Formula)** *A vessel contains 40 litres of milk. 4 litres are removed and replaced with water. This operation is repeated once more. How much milk remains?*

• Step 1: Fraction replaced k = 4/40 = 1/10. Number of operations n = 2.
• Step 2: Final milk = 40 × (1 − 1/10)² = 40 × (9/10)² = 40 × 81/100 = 32.4 litres.
• Answer: **32.4 litres of milk** remain.

**Example 3 (Milk-Water Concentration)** *Two vessels contain mixtures of milk and water. Vessel A has milk:water = 3:2 and vessel B has 5:3. If 10 litres from A and 8 litres from B are mixed, find the milk:water ratio in the new mixture.*

• Step 1: In A, milk fraction = 3/5, water = 2/5. In 10 L: milk = 6 L, water = 4 L.
• Step 2: In B, milk = 5/8, water = 3/8. In 8 L: milk = 5 L, water = 3 L.
• Step 3: Total milk = 6 + 5 = 11 L, total water = 4 + 3 = 7 L.
• Step 4: Ratio = 11 : 7.
• Answer: **11:7**.

Common Mistakes

1. **Swapping the differences**: Students write (M − C₁):(C₂ − M) instead of (C₂ − M):(M − C₁). The allegation cross-diagram prevents this—always put the *other* ingredient's difference opposite. **Fix**: Draw the diagram every time: mean in centre, higher value top-left, lower bottom-left, diagonal differences.

2. **Forgetting the exponent in replacement**: Applying (1 − k) only once when the operation repeats n times. For two replacements, it's (1−k)², not 2×(1−k). **Fix**: Memorise the formula with the exponent n. Write it out explicitly before substituting.

3. **Mixing up fraction and ratio**: After allegation gives 3:2, stating "3/2 of cheaper rice" instead of recognising it means 3 parts cheaper to 2 parts dearer. **Fix**: Ratio a:b means a/(a+b) fraction of total is the first component, b/(a+b) the second.

4. **Ignoring volume change**: Some problems state "on mixing, 2% volume is lost" or gained. Students apply allegation ignoring this. **Fix**: Read the problem. If volume changes, adjust final total before calculating concentrations.

5. **Unit mismatch**: Mixing ₹/kg price with ₹/quintal or litres with millilitres without converting. **Fix**: Convert everything to the same unit before starting calculations.

Quick Reference

• **Allegation ratio**: (Higher − Mean) : (Mean − Lower) gives ratio of quantities.
• **Replacement n times**: Remaining original = Initial × (1 − fraction)ⁿ.
• **Milk fraction in a:b ratio**: a/(a + b); water fraction: b/(a + b).
• **Pure substance mixing**: Pure ingredient has 100% concentration; use 100 or 0 as C₁ or C₂.
• **Three-component mix**: Apply allegation pairwise or solve algebraically—no single-step shortcut.
• **Cross-check with weighted average**: Final answer's mean should satisfy (Q₁V₁ + Q₂V₂)/(Q₁ + Q₂) = M.

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**Exam Tip**: Draw the allegation diagram neatly in rough work. It takes 5 seconds and eliminates sign errors. For replacement, write the formula first, then plug in numbers—don't try mental math with exponents.