Question

# A pot contains $82litres$ of pure milk. $\dfrac{1}{3}$ of the milk is replaced by the same amount of water. Again $\dfrac{1}{3}$ of the mixture is replaced by that amount of water. Find the ratio of milk and water in the new mixture.

Hint:Find out the one by third value in the $82litres$ of pure milk. Subtract that from the mixture quantity and add the same amount of water so that there will be no change in the total quantity at last. Now, again remove one third of the mixture but this time find out how much milk and water are removed according to their proportions and add the same amount of water. At last take the ratio of milk to water.

According to the problem given, they mentioned to us that in the initial stage there was only milk and the quantity of milk is $82litres$.
Later on, as per data one-third of the total milk is removed and the same amount of water is added.
So, firstly we should find the value of one-third in the $82litres$.
It can be simply found by dividing the value with the three.
So,
One-third of the milk $= \dfrac{1}{3} \times 82= 27.33liters$
And one more thing we need to remember is that in the question itself it mentioned that the same amount of water is added to the milk again.
So, the total quantity will not change at all.
Now, the mixture contains $82litres$ in which $27.33liters$ is water and rest is milk.
So, in order to find out the milk quantity simply we should remove the water quantity from the total mixture quantity.
Hence, milk quantity $= 82 - 27.33 = 54.67liters$
Up to now, everything is perfect. Now it is mentioned in the given data that again one-third of the present mixture is replaced with the same amount of water. So, as the total quantity of mixture is $82litres$, its one- third quantity is $27.33liters$ as we have already calculated before.
But, in this case this $27.33liters$ of taken out mixture contains both milk and water.
And they will be also in the one-third proportions with their respective quantities.
So, one-third in the $54.67liters$ of milk is $= \dfrac{1}{3} \times 54.67$
$= 18.22liters$
After removal of $18.22liters$ milk, remaining milk $= 54.67 - 18.22$
$= 36.45liters$
Similarly, we can find water too.
So, one-third in the $27.33liters$ of water is $= \dfrac{1}{3} \times 27.33= 9.11liters$
After removal of $9.11liters$ water, remaining water $= 27.33 - 9.11$
$= 18.22liters$
Now, we got both the quantities values of milk and water.
Milk and water are $36.45liters$ and $18.22liters$ respectively in the mixture.
Ratio of milk and water in the mixture $= \dfrac{{36.45}}{{18.22}}$
$\approx 2$
Hence, we can conclude that the ratio of milk and water in the mixture is $2:1$.

Note:This type of problem may go wrong when you calculate how much each liquid is removed when some amount of mixture is removed. To be perfect in your calculation you must check the proportions of the liquid in the mixture before removing and the removed quantity must contain the same proportions.