Mixtures & Alligations • Topic 4 of 4

Repeated Replacement

This is the single highest-frequency mixture pattern in CAT: from a vessel of volume V containing a pure substance, you draw off x units of the mixture and refill with x units of water — and you repeat this n times. After every operation the pure substance keeps the same fraction it had, so each draw removes the same proportion. The final pure quantity follows the geometric law final = initial × (1 − x/V)ⁿ. Equivalently, the fraction of pure substance remaining is (1 − x/V)ⁿ, and pure : original = (V − x)ⁿ : Vⁿ. The classic trap is mixing up "x drawn each time" with "total x drawn" — the formula needs the amount drawn in EACH single operation, with V being the (constant) total volume. Because the volume is restored to V after every refill, x/V stays fixed throughout. When the question gives the final pure amount and asks for n, take logs or test small integers; CAT values are almost always clean.

✅ Solved examples

1. A 80-litre vessel is full of milk. 8 litres is drawn and replaced by water; this is done twice. How much milk remains?
final = 80 × (1 − 8/80)² = 80 × (0.9)² = 80 × 0.81 = 64.8 litres of milk.
2. From 100 L of pure spirit, 20 L is drawn and replaced with water three times. Find the spirit left.
final = 100 × (1 − 20/100)³ = 100 × (0.8)³ = 100 × 0.512 = 51.2 litres.
3. A cask of 40 L wine has 4 L drawn and replaced by water four times. What fraction of wine remains?
fraction = (1 − 4/40)⁴ = (0.9)⁴ = 0.6561, i.e. 6561/10000 of the original wine.
4. A vessel of 50 L milk loses 5 L (replaced by water) each operation. After how many operations is the milk reduced to 32.805 L?
50 × (0.9)ⁿ = 32.805 ⇒ (0.9)ⁿ = 0.6561 = (0.9)⁴ ⇒ n = 4 operations.

✏️ Practice — try these, take hints as needed

1. 60 L pure milk; 6 L drawn and replaced by water twice. Milk left?
Use final = V(1 − x/V)ⁿ.
60 × (1 − 6/60)².
60 × (0.9)².
48.6 litres
2. From 100 L of wine, 10 L is removed and refilled with water 3 times. Wine remaining?
(1 − 10/100)³.
100 × (0.9)³.
100 × 0.729.
72.9 litres
3. A 64 L vessel of milk; 16 L drawn and replaced by water twice. Final milk?
x/V = 16/64 = 1/4.
64 × (3/4)².
64 × 9/16.
36 litres
4. After 2 equal draw-and-refill operations on a full vessel, milk : water = 9 : 16 by the end. What fraction was drawn each time?
Milk fraction left = 9/25.
(1 − x/V)² = 9/25.
1 − x/V = 3/5.
2/5 of the volume each time
5. 80 L of acid; 16 L drawn and replaced with water 3 times. How much WATER is now in the vessel?
Acid left = 80 × (1 − 16/80)³.
80 × (0.8)³ = 40.96 L acid.
Water = 80 − acid.
39.04 litres

📝 Topic test — 8 questions

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