Replacement and Repeated Dilution: The (1 − x/n)^k Formula

Replace and Repeat: Why This Formula Is a Gift

A vessel holds 40 litres of pure milk. A shopkeeper removes 4 litres and pours in 4 litres of water. He then does it again. How much milk is left? Most students subtract 4 + 4 from 40 and answer 32 litres. That is wrong, because the second 4 litres he removes is already a mixture and carries some water out with it.

The correct idea is that each round keeps the same fraction of the original liquid. Removing 4 from 40 keeps 36/40 = 9/10. Two rounds keep (9/10)² = 81/100, so milk left = 40 × 0.81 = 32.4 litres, and water = 7.6 litres.

This single pattern powers a whole family of mixture questions in SSC, Railway and BPSC TRE-4 papers. Learn the fraction idea once and you never need to build tables again.

1. A vessel holds 40 litres of pure milk. 4 litres are removed and replaced with water, and the process is repeated once more. How much milk is left?

2. A container has 80 litres of pure milk. 8 litres are drawn out and replaced with water, and this is done three times in all. How much milk remains?

3. A 60-litre mixture of milk and water is in the ratio 2 : 1. If 12 litres of the mixture are removed and replaced with water, how much milk is left?

4. In the previous mixture (60 litres, milk : water = 2 : 1, then 12 litres removed and replaced by water), what is the new ratio of milk to water?

5. A vessel has 64 litres of pure milk. A quantity is removed and replaced by water, and the same is done once more, leaving 36 litres of milk. What quantity was replaced each time?

6. A dealer mixes water with milk and sells the mixture at the cost price of pure milk, earning a 20% gain. How much water is there in one litre of the mixture?

7. A tank holds 1000 litres of pure milk. Ten percent of the liquid is drawn off and replaced with water, and this is done three times. How much water is in the final mixture?

8. A vessel of capacity n litres contains pure liquid. If x litres are replaced by another liquid and the process is done k times, the fraction of the original liquid left is: