The Rule of Alligation (Cross Method)

🎯 Learning Objective

Find the ratio in which two ingredients at different prices or strengths must be mixed to hit a target mean value.

💡 Concept

• Alligation finds the ratio for mixing two things of different values to reach a required average (mean) value.
• Rule: (Quantity of cheaper) / (Quantity of dearer) = (Dearer price − Mean) / (Mean − Cheaper price)
• Cross method: write cheaper and dearer values at top corners, mean in the middle, and subtract diagonally.
• The mean value always lies BETWEEN the two ingredient values — never outside.
• It works for price, concentration, percentage, or speed — any average of two groups.

🧮 Key Formulas

Cheaper qty : Dearer qty = (Dearer − Mean) : (Mean − Cheaper)

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Mean = (C₁·Q₁ + C₂·Q₂) / (Q₁ + Q₂)

✏️ Easy Example

Q. Rice at ₹30/kg is mixed with rice at ₹40/kg to make a mixture worth ₹34/kg. Find the ratio.

1. Cheaper = 30, Dearer = 40, Mean = 34
2. Ratio = (40 − 34) : (34 − 30) = 6 : 4
3. Simplify 6 : 4 = 3 : 2

Answer: 3 : 2 (cheaper : dearer)

🇮🇳 Real-Life Example

A kirana shop mixes cheaper ₹30 rice with premium ₹40 rice to sell a ₹34 blend — the alligation ratio tells the shopkeeper exactly how many kilos of each to pour into the sack.

📝 Exam-Level Example

Q. In what ratio must rice at ₹25/kg and ₹35/kg be mixed to make 60 kg of mixture costing ₹28/kg? How much of each?

1. Ratio = (35 − 28) : (28 − 25) = 7 : 3
2. Total parts = 7 + 3 = 10, for 60 kg
3. Cheaper = 7/10 × 60 = 42 kg
4. Dearer = 3/10 × 60 = 18 kg

Answer: Ratio 7 : 3 → 42 kg of ₹25 rice and 18 kg of ₹35 rice

📝 Exam-Level Example

Q. In what ratio should water (free) be mixed with milk worth ₹12/litre so the mixture is worth ₹8/litre?

1. Water is free, so its value = ₹0 (cheaper). Milk = ₹12 (dearer). Mean = ₹8
2. Ratio water : milk = (12 − 8) : (8 − 0) = 4 : 8
3. Simplify = 1 : 2

Answer: Water : Milk = 1 : 2

🪄 Memory Trick

Water is always the 'free' ingredient with value 0. Put it as the cheaper corner and the ratio pops out instantly.

⚠️ Common Mistakes

• ❌ Placing the mean outside the two values (impossible — it must be in between)
• ❌ Mixing up which side of the ratio is cheaper vs dearer
• ❌ Forgetting to simplify the final ratio (6 : 4 should become 3 : 2)

🏆 Exam Tips

• ✅ Always write the SMALLER value on the left so cheaper : dearer stays consistent
• ✅ Alligation only handles TWO groups at a time — for three, combine two first

📌 Summary

• Cheaper : Dearer = (Dearer − Mean) : (Mean − Cheaper)
• Mean always lies between the two values
• Water counts as value 0
• Simplify the ratio at the end