# Difficult Alligations or Mixtures Solved QuestionAptitude Discussion

 Q. 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 removals and replacement, what is the ratio of milk and water in the resultant mixture?
 ✖ A. 17 : 3 ✔ B. 9 : 1 ✖ C. 3 : 17 ✖ D. 5 : 3

Solution:
Option(B) is correct

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 $W_o$ be the amount of water in the mixture originally = 8 litres.
Let $W_r$ be the amount of water in the mixture after the replacements have taken place.

Then, $\dfrac{W_r}{W_o}=\left(1-\dfrac{R}{M}\right)^n$
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, $\dfrac{W_r}{W_o}=\left(\dfrac{1}{2}\right)^2=\dfrac{1}{4}$

Therefore, $W_r = \dfrac{W_o}{4}=\dfrac{8}{4}=2$ litres

Hence, the mixture will have 18 litres of milk and 2 litres of water.

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

## (5) Comment(s)

Prachi
()

Quantity removed =10/20=1/2

Water initialy=8

Water after 2 replacement= 8 * 1/2 * 1/2 =2

Therefore milk =20-2 =18

Akshay
()

$ax^2+bx+c=0$

Okay i am testing latex. Please dont mind

Priya
()

plz explain clearly i m not getting

Raaj
()

In step 2 how it get 18 li of milk n 2 li of water

Ruby
()

after first step the milk and water in the mixture is 16 litres and 4 litres.

in step we have to remove 10 litres in the ratio of 3:2 (3 part of milk and 2 part of water)

in 10 litre 3 part is 6 litre and 2 part is 4 litre.

after removing 6ltr milk and 4 ltr water now we have 10 ltr milk and 2 ltr water

now we have to add 10 ltr milk to the mix.

so milk and water in the mix is (18,2)

ratio: 18:2 or 9:1