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Mixture & Alligation

Quantitative Aptitude / DI · Arithmetic Applications

Useful for: SSC CGL, Railway RRB, IBPS Banking, OSSC RHT — Regular High School Teacher, OSEPA Primary Teacher / JT Category-I, OSEPA Upper Primary — Science, OSSC Mathematics, OSSSC Govt Physical Education Teacher (PET)

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Overview

Alligation finds the mixing ratio from two prices and the mean price; partnership shares profit in proportion to capital × time.

Key points

Alligation: quantity of cheaper : quantity of dearer = (dearer − mean) : (mean − cheaper).
Repeated replacement: final pure quantity = initial × (1 − x/total)^n after n replacements of x units.
Partnership: profit ratio = capital × time for each partner.
A working partner’s salary is paid first; the rest is shared in the investment ratio.
Alligation works for any mean: prices, percentages, speeds, marks.

Explanation

For three mixtures, combine two first, then the third.
Concentration mix: c = (c1 × q1 + c2 × q2)/(q1 + q2).
If capital changes mid-year, split the year into periods.
Water is a "zero-price" ingredient in milk-water alligation.

Step by step

Write the two values and the mean.
Apply the alligation cross or partnership ratio.
Scale to the required quantity.

Examples

Rice at ₹30/kg and ₹40/kg mixed to cost ₹36/kg: (40 − 36):(36 − 30) = 4:6 = 2:3 (cheaper : dearer).
A invests ₹5,000 for 12 months, B ₹6,000 for 10 months → 60,000 : 60,000 = 1:1.
From 40 L of milk, 4 L is replaced by water twice: milk left = 40 × (1 − 4/40)² = 32.4 L.

Traps and advanced points

Trap: Reversing the alligation ratio (cheaper ↔ dearer).
Trap: Ignoring time in partnership.
Trap: Subtracting the same 4 L each time instead of using the replacement formula.

Quick revision

(D − M):(M − C).
Capital × time.
Replacement: (1 − x/T)^n.

Practice MCQs

Two solutions of 36% and 57% concentration are mixed in ratio 1:2. Find concentration of mixture.
1. 55%
2. 50%
3. 45%
4. 60%
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Answer: (b) 50%. Use weighted average.
Two solutions of 38% and 66% concentration are mixed in ratio 2:3. Find concentration of mixture.
1. 64.8%
2. 59.8%
3. 54.8%
4. 49.8%
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Answer: (c) 54.8%. Use weighted average.
Two solutions of 16% and 42% concentration are mixed in ratio 1:1. Find concentration of mixture.
1. 39%
2. 24%
3. 29%
4. 34%
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Answer: (c) 29%. Use weighted average.
Two solutions of 19% and 43% concentration are mixed in ratio 2:3. Find concentration of mixture.
1. 28.4%
2. 38.4%
3. 43.4%
4. 33.4%
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Answer: (d) 33.4%. Use weighted average.
Two solutions of 22% and 44% concentration are mixed in ratio 1:1. Find concentration of mixture.
1. 28%
2. 43%
3. 33%
4. 38%
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Answer: (c) 33%. Use weighted average.
Two solutions of 38% and 61% concentration are mixed in ratio 3:2. Find concentration of mixture.
1. 47.2%
2. 42.2%
3. 57.2%
4. 52.2%
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Answer: (a) 47.2%. Use weighted average.
Two solutions of 32% and 59% concentration are mixed in ratio 1:2. Find concentration of mixture.
1. 60%
2. 55%
3. 50%
4. 45%
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Answer: (c) 50%. Use weighted average.
Two solutions of 27% and 49% concentration are mixed in ratio 2:3. Find concentration of mixture.
1. 35.2%
2. 40.2%
3. 45.2%
4. 50.2%
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Answer: (b) 40.2%. Use weighted average.
Two solutions of 18% and 46% concentration are mixed in ratio 2:3. Find concentration of mixture.
1. 44.8%
2. 29.8%
3. 34.8%
4. 39.8%
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Answer: (c) 34.8%. Use weighted average.
Two solutions of 19% and 48% concentration are mixed in ratio 1:2. Find concentration of mixture.
1. 33.33%
2. 48.33%
3. 38.33%
4. 43.33%
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Answer: (c) 38.33%. Use weighted average.
Two solutions of 31% and 57% concentration are mixed in ratio 1:1. Find concentration of mixture.
1. 49%
2. 44%
3. 39%
4. 54%
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Answer: (b) 44%. Use weighted average.
Two solutions of 32% and 54% concentration are mixed in ratio 2:3. Find concentration of mixture.
1. 40.2%
2. 50.2%
3. 45.2%
4. 55.2%
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Answer: (c) 45.2%. Use weighted average.
Two solutions of 38% and 66% concentration are mixed in ratio 1:1. Find concentration of mixture.
1. 62%
2. 47%
3. 57%
4. 52%
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Answer: (d) 52%. Use weighted average.
Two solutions of 36% and 62% concentration are mixed in ratio 2:3. Find concentration of mixture.
1. 56.6%
2. 61.6%
3. 46.6%
4. 51.6%
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Answer: (d) 51.6%. Use weighted average.
Two solutions of 24% and 48% concentration are mixed in ratio 2:3. Find concentration of mixture.
1. 43.4%
2. 38.4%
3. 48.4%
4. 33.4%
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Answer: (b) 38.4%. Use weighted average.