Average, Mixture & Alligation • Topic 4 of 4

Mixtures & Replacement

When a fraction of a mixture is repeatedly removed and replaced with one ingredient (the classic 'milk-and-water, remove r litres and top up with water' problem), the remaining original ingredient after n operations is initial x (1 - r/V)^n, where V is the total volume. This geometric decay is the SSC favourite for liquid replacement.

✅ Solved examples

1. A 40 L cask of milk: 8 L removed and replaced with water, once. Milk left?
40 x (1 - 8/40) = 40 x 4/5 = 32 L.
2. Same cask, the operation done twice. Milk left?
40 x (4/5)^2 = 40 x 16/25 = 25.6 L.
3. A vessel has 50 L milk; 10 L drawn and replaced by water twice. Milk:water now?
Milk = 50 x (4/5)^2 = 32; water = 18; ratio 32:18 = 16:9.
4. 20% of wine is drawn and replaced with water, repeated. After how many steps is wine below 50%?
Each step x 0.8: after 1 -> 80%, 2 -> 64%, 3 -> 51.2%, 4 -> 40.96%. So after 4 operations.

✏️ Practice — try these, take hints as needed

1. 60 L milk, remove 12 and top with water once. Milk left?
60 x (1 - 12/60).
60 x 4/5.
—
48 L
2. 80 L, remove 16 and replace twice. Milk left?
80 x (4/5)^2.
80 x 16/25.
—
51.2 L
3. 25 L, remove 5 and replace twice. Milk:water?
Milk = 25 x (4/5)^2 = 16.
Water = 9.
—
16:9
4. 100 L, remove 20 and replace once. Milk %?
100 x 0.8.
—
—
80%
5. 90 L, remove 30 and replace once. Milk left?
90 x (1 - 30/90).
90 x 2/3.
—
60 L

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Alligation Rule Take the chapter test →