Mixtures Question: Remove & Replace

This aptitude practice question is a mixtures and alligation question. Concept tested: Remove and replace it with one of the components of a mixture repeatedly. Solved using the formula.

Question

A 20 litre mixture of milk and water contains milk and water in the ratio 3 : 2. 10 litres of the mixture is removed and replaced with pure milk and the operation is repeated once more. At the end of the two removal and replacement, what is the ratio of milk and water in the resultant mixture?

1. 17 : 3
2. 9 : 1
3. 3 : 17
4. 5 : 3

Correct Answer      Choice (2). 9 : 1

The 20 litre mixture contains milk and water in the ratio of 3 : 2. Therefore, the mixture contains 12 litres of milk and 8 litres of water.

Step 1. When 10 litres of the mixture is removed, 6 litres of milk is removed and 4 litres of water is removed.
Therefore, there will be 6 litres of milk and 4 litres of water left in the container.
It is then replaced with pure milk of 10 litres. Now the container will have 16 litres of milk and 4 litres of water.

Step 2. When 10 litres of the new mixture is removed, 8 litres of milk and 2 litres of water is removed.
The container will have 8 litres of milk and 2 litres of water in it.
Now 10 litres of pure milk is added. Therefore, the container will have 18 litres of milk and 2 litres of water in it at the end of the second step.

Therefore, the ratio of milk and water is 18 : 2 or 9 : 1.

Formula Method
We are essentially replacing water in the mixture with pure milk.
Let WO be the amount of water in the mixture originally = 8 litres.
Let WR be the amount of water in the mixture after the replacements have taken place.
Then, $$frac {{W}_{R}} {{{W}_{O}}}$ = $$1-$frac{R}{M}{$}^{n}), where R is the amount of the mixture replaced by milk in each of the steps, M is the total volume of the mixture and n is the number of times the cycle is repeated. Hence, $\frac {{W}_{R}} {{{W}_{O}}}$ = $$1-$frac{10}{20}{$}^{2}) = $$$frac{1}{2}{$}^{2}) = $\frac{1}{4}$ Therefore, WR = $\frac {{W}_{O}} {4}$ = $\frac{8}{4}$ = 2 litres. WR is the quantity of water in the mixture after the process is done twice. So, the final quantity of water in the 20-litre mixtures is 2 litres. Hence, the mixture will have 18 litres of milk and 2 litres of water. The correct answer is Choice$2).

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