Mixtures and Alligations - Quant/Math - CAT 2008
Question 4 the day: July
3, 2002
- 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 - (2)
Solution:
The 20 litre mixture contains milk and water in the ratio of 3 : 2. Therefore,
there will be 12 litres of milk in the mixture and 8 litres of water in the
mixture.
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.
Shortcut.
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, Wr/Wo=
(1-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,Wr/Wo
=1/2 ^ 2
=1/4
Therefore, WR =Wo/4=8/4
= 2 litres.
Hence the mixture will have 18 litres of milk and 2 litres of water.
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