← All topics🔁 Removal & ReplacementMaths
Mixture & Alligation · Topic 2 of 6

🔁 Removal & Replacement

3 exam question types, fully solved

A classic SSC trap: draw some liquid out of a full vessel, top it up with water, and repeat. Each round the original liquid gets a little weaker — but it does so by the same fraction every time, which gives one clean formula.

The dilution formula

Why the same fraction each round? Because whatever you scoop out is a sample of the current mixture, not just the original liquid. After rounds of removing from a vessel of volume :

So the leftover fraction is , and the water is whatever is left over: .

• To find the fraction replaced when only water is added, track the OTHER liquid (the fixed one): final proportion initial proportion , where is the fraction drawn off.

• Single swap, new ratio: remove the chunk in the current ratio, then add the new liquid — re-read the amounts.

⚖️Alligation LabCross rule & replacement
Vessel (V) litres
Replaced (x) litres
Operations (n)
100%
start
80%
op 1
64%
op 2
milk (original)water added
Each round keeps the fraction (1 − x/V) of whatever milk is present: Milk left = V × (1 − x/V)ⁿ = 40 × (1 − 8/40)^2 = 40 × (0.8)^2 = 25.6 L
Milk left = 25.6 L ·  water = 14.4 L

From 100 L of milk, 10 L is removed and replaced with water, done twice. Milk left L.

📝Practice Questions

Q1A 80-litre cask of milk has 20 litres drawn and replaced with water, done twice. The milk left is:

Q2A vessel of pure juice has 1/5 drawn off and replaced with water, repeated once more. The fraction of juice left is:

Q3Milk and water are in the ratio 2:3. What fraction must be replaced with water to make it 1:4? (track the milk)

📚

Real exam questions — Removal & Replacement

3 question types · 9 solved examples from real SSC papers

Draw out, top up, repeat — the original liquid shrinks by the same fraction each round. One formula, final = initial × (1 − x/V)ⁿ, handles it all.

How to solve this type
When you draw out xx litres from a vessel of VV litres and top up with water, the original liquid keeps shrinking by the SAME fraction each round, because what you remove is a sample of whatever is currently inside. After nn rounds:
Original left =V×(1−xV)n=V\times\left(1-\dfrac{x}{V}\right)^n.
The leftover fraction is (1−xV)n\left(1-\dfrac{x}{V}\right)^n. Compute (1−xV)\left(1-\dfrac{x}{V}\right) once, raise it to the power nn, then multiply by VV. Water =V−=V- (original left).

From a cask of 64 litres of milk, 8 litres are drawn out and replaced with water. The same is done a second and a third time. Find the final ratio of milk to water.

A343 : 179B333 : 169C443 : 179D343 : 169

A cistern has 100 litres of pure syrup. 10 litres are drawn out and replaced with water; this is repeated once more. Find the final ratio of syrup to water.

A18 : 1B89 : 14C81 : 19D18 : 17

A container has 100 litres of acid and water in the ratio 4 : 1. 20 litres is removed and replaced with water; then 20 litres is removed again and replaced with pure acid. Find the final percentage of acid.

A76.8%B78.4%C81.6%D71.2%