मिश्रण और पृथ्थीकरण

हिंदी में पढ़ें

Mixture and Alligation (मिश्रण और एलिगेशन)

Learning Goals

By the end of this chapter you will be able to:

1. Find the average price or strength of a mixture.
2. Use the alligation cross to find the ratio in which two things must be mixed.
3. Solve repeated replacement questions (milk taken out, water added).
4. Find the new ratio when one component is added to a mixture.
5. Apply alligation to any "weighted average" problem — prices, speeds, marks, profits.

The Hook: The Chai Stall Puzzle

A chai stall owner mixes tea leaves costing ₹200 per kg with leaves costing ₹300 per kg. He wants the blend to cost exactly ₹240 per kg. How much of each should he use? You could set up equations — or you could draw a small cross and get the answer in 10 seconds. That cross is alligation.

Why Students Struggle

• They take the simple average of two prices even when the quantities are different.
• They write the alligation ratio upside down.
• They subtract the same amount every time in replacement questions.

🧠 Active Recall: If equal amounts of ₹200 and ₹300 tea are mixed, what is the price of the mixture? What if the amounts are not equal?

Chunk 1: Mean Price (Weighted Average)

Mean price = (q₁ × c₁ + q₂ × c₂) ÷ (q₁ + q₂)

Example: 3 kg at ₹40 and 2 kg at ₹50 → (120 + 100) ÷ 5 = ₹44 per kg.

Chunk 2: The Alligation Cross

   cheaper (c)          dearer (d)
            \          /
             mean (m)
            /          \
     (d − m)            (m − c)

Cheaper : Dearer = (d − m) : (m − c)

Chai example: c = 200, d = 300, m = 240 → (300 − 240) : (240 − 200) = 60 : 40 = 3 : 2.

Mnemonic — "Cross and Subtract, Far goes Near": each quantity is the distance of the *other* price from the mean.

🧠 Active Recall: In the chai example, why does the cheaper tea get the bigger share?

Chunk 3: Replacement (Removal and Refilling)

A vessel has V litres of milk. x litres are taken out and replaced by water, n times.

Milk left = V × (1 − x/V)ⁿ

Example: 80 litres of milk, 20 litres replaced twice → 80 × (3/4)² = 45 litres (not 40).

Chunk 4: Adding One Component

Milk : water = 4 : 1 in 50 litres → milk 40, water 10. Add 10 litres of water → 40 : 20 = 2 : 1.

Chunk 5: Alligation Beyond Mixtures

Any average of two groups works the same way: average speeds, average marks of two sections, profit on two parts of a sale.

Summary Table

📅 Review this chapter after 2 days, then solve one practice set with a timer.

⚠️ Caution: Examiner's Trap

1. Upside-down ratio: the option with the ratio reversed is always present. The cheaper item's share is (dearer − mean).
2. Linear replacement: subtracting x litres of milk each time ignores that later removals contain water too.
3. Simple average: averaging two concentrations without weighting by volume.