Mixture & Alligation — Advanced Exam Problems

🎯 Learning Objective

Push alligation into hard territory — averages, profit-loaded mixtures, ratio reversal and two-vessel blends.

💡 Concept

• Alligation beyond prices: class averages, wages, speeds — anything with a weighted mean
• Profit twist: if the mixture is SOLD, divide SP by (1 + profit/100) to get the mean COST first
• Changing a ratio by draw-and-replace: track the pure component — only the drawn mixture removes it
• Blending two mixtures: the cross works on the FRACTION of one ingredient, never on raw ratios
• Always close with a verification line — recompute the final mixture from your answer

🧮 Key Formulas

Mixture CP for the cross = SP ÷ (1 + profit/100)

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Milk removed on drawing x litres = x × (milk fraction of the vessel)

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Blend cross: Q_A : Q_B = (mean − f_B) : (f_A − mean)

✏️ Easy Example

Q. In what ratio must tea at ₹62/kg be mixed with tea at ₹72/kg so that the mixture, sold at ₹77/kg, gives a 10% profit?

1. Selling at ₹77 with 10% profit → the mixture's COST must be 77/1.1 = ₹70 — alligation always works on cost, never on SP
2. Cross method: cheaper 62, dearer 72, mean 70
3. Ratio = (72 − 70) : (70 − 62) = 2 : 8
4. Simplify → 1 : 4 (cheaper : dearer)

Answer: 1 : 4

🇮🇳 Real-Life Example

Blending two coffee batches to hit a target strength, or HR balancing skilled-unskilled wage budgets — one cross diagram does both.

📝 Exam-Level Example

Q. In a factory the average daily wage of all 150 workers is ₹280. Skilled workers average ₹340 and unskilled workers ₹240. Find the number of skilled workers.

1. The overall average ₹280 sits between ₹240 and ₹340 — so alligation applies to people, not just prices
2. Skilled : Unskilled = (280 − 240) : (340 − 280) = 40 : 60 — each group takes the gap on the OPPOSITE side of the mean
3. = 2 : 3, so out of every 5 workers, 2 are skilled
4. Skilled = (2/5) × 150 = 60
5. Check: 60 × 340 + 90 × 240 = 20400 + 21600 = 42000, and 42000/150 = 280 ✓

Answer: 60 skilled workers

📝 Exam-Level Example

Q. A vessel contains 60 L of milk and water in the ratio 2 : 1. How much of the mixture must be drawn off and replaced with water so that the ratio becomes 1 : 2?

1. Now: milk = (2/3) × 60 = 40 L, water = 20 L; target 1 : 2 in the same 60 L means milk must fall to 20 L
2. Water poured in later cannot remove milk — milk leaves ONLY inside the drawn-out mixture
3. Every litre drawn carries 2/3 litre milk (the current ratio), so drawing x litres removes 2x/3 of milk
4. Set up: 40 − 2x/3 = 20 → 2x/3 = 20
5. x = 30 — draw 30 L of mixture and pour back 30 L of water; check: milk 20, water 10 + 30 = 40 → 1 : 2 ✓

Answer: 30 litres

📝 Exam-Level Example

Q. Two vessels A and B contain spirit and water in the ratios 5 : 2 and 7 : 6. In what ratio must they be mixed to get a mixture with spirit : water = 8 : 5?

1. Convert every ratio to a spirit FRACTION: A = 5/7, B = 7/13, target = 8/13 — raw ratios cannot enter the cross
2. Alligation: A : B = (target − B) : (A − target) = (8/13 − 7/13) : (5/7 − 8/13)
3. First gap = 1/13; second gap = (65 − 56)/91 = 9/91 — bring 5/7 and 8/13 to the common denominator 91
4. A : B = 1/13 : 9/91 = 7/91 : 9/91 = 7 : 9
5. Check: 7 L of A gives 5 L spirit; 9 L of B gives 63/13 L; spirit total = 128/13, water = 80/13 → 128 : 80 = 8 : 5 ✓

Answer: 7 : 9

🪄 Memory Trick

Whatever the story — wages, marks, milk — convert each group to ONE number (fraction or average), put the mean in the centre, subtract diagonally. One diagram runs the whole chapter.

⚠️ Common Mistakes

• ❌ Feeding the selling price into the cross — alligation runs on cost, so strip the profit first
• ❌ Using ratios (5 : 2, 7 : 6) directly in the cross instead of fractions 5/7 and 7/13
• ❌ Letting added water 'remove' milk — only the drawn-out mixture carries milk away

🏆 Exam Tips

• ✅ Alligation answers sit cross-wise: each group's quantity equals the FAR gap
• ✅ Bring fractions to one denominator (65/91 vs 56/91) — the gaps become error-proof
• ✅ Verify the blend by recomputing the final ratio — 30 seconds for guaranteed marks

📌 Summary

• Alligation works on ANY average — wages, marks, concentrations
• Strip profit to reach cost before the cross
• Ratio reversal: only drawn mixture removes the pure part
• Two-mixture blends → cross on fractions, verify at the end