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 removals and replacement, what is the ratio of milk and water in the resultant mixture?

Answer 1

Let $W_o$ be the amount of water in the mixture with pure milk$=8l$

Let $W_r$ be the amount of water in the mixture after the replacement have taken place.

$\Rightarrow \frac{W_r}{W_o}=(1-\frac{R}{M}{)}^n$

Where R is the amount of 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, $\frac{W_r}{W_o}=(\frac{1}{2}{)}^2=\frac{1}{4}$

$\therefore W_r=\frac{W_o}{4}=\frac{8}{4}=2l$

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

$\therefore$ ratio of $milk$ and $water=18:2=9:1$