मिश्रण और पृथ्थीकरण

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Mixture & Alligation

Chapter Outline & Learning Goals

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

1. Understand what a mixture is and how ratios behave inside it.
2. Master the Rule of Alligation — the fastest cross-method for two-component mixtures.
3. Solve replacement problems (the classic "milk drawn out and water added" type) using a single formula.
4. Handle mixing of mixtures, price-based mixtures, and three-component extensions.
5. Spot the traps examiners set in SSC CGL Tier-II, IBPS PO Mains, CAT and CSAT.

The Hook: The Milkman Who Never Studied Maths

A milkman in your neighbourhood mixes water into milk every single day — and yet he never calculates wrongly. Ask him how, and he'll say: *"I just know how much profit I want."*

That is exactly what Alligation is. It is not a formula invented by mathematicians; it is the street-smart shortcut that shopkeepers, goldsmiths and chemists have used for centuries. The moment you see the "cross diagram," you will realise that questions which look like 2-minute algebra problems are actually 10-second ratio problems.

And here is the uncomfortable truth: most aspirants lose marks in this chapter not because it is hard, but because they solve it with equations when a single diagonal subtraction would do. Let's fix that — permanently.

1. What Is a Mixture, Really?

A mixture is simply two or more quantities combined together — milk and water, two varieties of rice, two alcohol solutions, two alloys of gold and copper.

The key insight: in a mixture, only ratios matter, not absolute quantities. A 3:1 milk-water mixture behaves identically whether it is 4 litres or 400 litres.

Alligation is the rule that tells us the ratio in which two ingredients (at two different "values" — price, concentration, percentage) must be mixed to get a mixture of a desired mean value.

🧠 Active Recall: Why do we say "only ratios matter" in a mixture? If a mixture has milk and water in 5:2, what fraction of the mixture is water?

2. The Rule of Alligation — The Cross That Solves Everything

Suppose ingredient A has value a (cheaper) and ingredient B has value b (dearer), and we mix them to get a mean value m (where a < m < b). Then:

        a              b
          \          /
            \  m  /
            /      \
          /          \
     (b − m)      (m − a)

Quantity of A : Quantity of B = (b − m) : (m − a)

Notice the diagonal subtraction: the cheaper ingredient's quantity comes from the *dearer* value minus the mean, and vice versa. The values "cross over" — hence the name *alligation* (from Latin *alligare*, "to bind together").

Example: Rice at ₹40/kg and rice at ₹60/kg are mixed to get rice worth ₹48/kg. Ratio?

• (60 − 48) : (48 − 40) = 12 : 8 = 3 : 2

Ten seconds. No equations.

🧠 Active Recall: In the cross diagram, why does the cheaper ingredient's quantity come from (b − m) and not (m − a)? What would go wrong if you didn't cross?

3. Why Does the Cross Work? (The One-Minute Proof)

Let quantities be x (at value a) and y (at value b). The mean is:

m = (ax + by) / (x + y)

Cross-multiply: mx + my = ax + by
⟹ x(m − a) = y(b − m)
⟹ x : y = (b − m) : (m − a)

That's it. The cross diagram is just this equation drawn as a picture. Once you trust the picture, you never need the equation again.

Mnemonic — "CROSS to WIN":
> Cheaper quantity comes from the Remote (far) Opposite Side Subtraction.

Or simply remember: *"Far value gives near quantity"* — the quantity of each ingredient is found from the value on the opposite end of the cross.

🧠 Active Recall: Derive the alligation ratio from m = (ax + by)/(x + y) without looking above. Can you do it in under a minute?

4. Concentration Problems — Alligation Beyond Money

Alligation works for any mean value: price, percentage of alcohol, profit %, speed, even average marks.

Example: A 20% alcohol solution is mixed with a 50% alcohol solution to get a 35% solution. Ratio?

• (50 − 35) : (35 − 20) = 15 : 15 = 1 : 1

Trap alert: If the question asks "how much of the 50% solution is needed to make 60 L of the final mixture," don't stop at the ratio. Ratio 1:1 in 60 L means 30 L. Advanced exams (CAT, PO Mains) always add this second step.

5. The Replacement Formula — The Crown Jewel of This Chapter

This is the most-tested pattern in SSC and Banking exams.

Setup: A vessel contains V litres of pure liquid (say milk). x litres are drawn out and replaced with water. This is repeated n times.

Final quantity of pure milk:

$$\text{Milk left} = V\left(1 - \frac{x}{V}\right)^n$$

Example: A 40 L vessel of pure milk; 8 L is replaced by water, twice.

• Milk left = 40 × (1 − 8/40)² = 40 × (4/5)² = 40 × 16/25 = 25.6 L

Why it works: Each operation removes the *same fraction* (x/V) of whatever milk is present — so milk shrinks by a constant factor each time, exactly like compound depreciation.

Memory trick: Think of it as "Compound Interest in Reverse" — the same (1 − r)ⁿ structure you already know, where r = x/V.

🧠 Active Recall: In the replacement formula, what plays the role of the "rate"? If 1/5 of the mixture is replaced each time, what fraction of milk remains after 3 operations?

6. Variations of Replacement (Where Toppers Separate Themselves)

Variation 1 — Replacement with another liquid, then asked for ratio:
After finding milk left, water = V − milk left. Ratio follows directly.

Variation 2 — Different amounts replaced each time:
Multiply the factors individually: Milk left = V × (1 − x₁/V)(1 − x₂/V)(1 − x₃/V)

Variation 3 — The vessel starts as a mixture, not pure liquid:
Apply the factor only to the component asked. If milk is initially 60% of V, then milk left = 0.6V × (1 − x/V)ⁿ.

Variation 4 — "After n operations the ratio becomes a:b, find n":
Set V(1 − x/V)ⁿ / V = a/(a+b) and solve by trial with small n (exams keep n = 2 or 3).

7. Mixing Two Mixtures

When two *mixtures* (not pure ingredients) are combined, work component by component.

Example: Vessel A has milk:water = 3:2; Vessel B has milk:water = 7:3. Equal quantities are mixed. Find the new ratio.

Take 10 L from each (LCM-friendly choice):

• Milk = 10×(3/5) + 10×(7/10) = 6 + 7 = 13
• Water = 4 + 3 = 7
• New ratio = 13 : 7

Shortcut via alligation: Treat milk-fractions 3/5 and 7/10 as the two "values," and the desired mean as the unknown — alligation works here too if the final ratio is given instead.

🧠 Active Recall: When mixing two mixtures in equal amounts, why can we pick any convenient quantity (like 10 L) from each? What changes if the quantities are unequal?

8. Three Ingredients and Repeated Alligation

Alligation is a two-ingredient tool. For three ingredients, apply it in stages: mix two first, then mix the result with the third. Or use the weighted-average equation directly.

Example pattern (CAT-level): Three teas at ₹120, ₹150 and ₹200 per kg must give a blend at ₹160/kg, with the first two in ratio 2:3.

• Step 1: Blend of first two at 2:3 → value = (2×120 + 3×150)/5 = ₹138
• Step 2: Alligate ₹138 with ₹200 for mean ₹160 → (200−160):(160−138) = 40:22 = 20:11
• Final ratio = 2 : 3 : (5 × 11/20) → scale to integers: 8 : 12 : 11

9. Examiner's Traps — Read This Twice

1. "Water is free" trap: Mixing water (₹0) with milk — alligation still works; just use 0 as the cheaper value.
2. Profit-based mixture: If milk at ₹40/L is mixed with water and sold at ₹48/L with 20% profit, the *cost price of the mixture* is 48/1.2 = ₹40... wait — check: CP = 48 × 100/120 = ₹40. Then alligate 40 (milk) and 0 (water) against mean 40 → pure milk! Questions like this test whether you convert SP to CP *before* alligating.
3. Ratio vs. quantity confusion: "In what ratio" and "how many litres" are different questions — the second needs the total.
4. Replacement trap: The formula gives *milk left*, but the question may ask for *water added* or the *final ratio*. Always finish the last step.
5. Percentage vs. fraction: 20% alcohol means alcohol fraction 1/5 — keep one consistent form throughout.

🧠 Active Recall: In a profit-based mixture problem, what must you compute *before* applying alligation — and why can't you use the selling price directly?

10. Summary Table

📌 Review Reminder

Review this chapter after 2 days. Redo the replacement formula derivation and solve one alligation problem from memory. Spaced repetition is what converts this chapter from "understood" to "unshakable" in the exam hall.