Question

From a cask of milk, containing 40 litre, 8 litres are drawn out and the cask is filled up with water. If the same process is repeated a second, then a third time, what will be the number of litres of milk left in the cask.
A.20.48
B.20
C.19.48
D.21.58

Answer 1

Hint: To find the number of litres of milk left in the cask, we will try to chart out the process and we get: Initially 40 litre of milk is there in the cask when we replace it with 8 litre of water and we are left with 32 litres of milk. Again, when we replace, we calculate the ratio of milk to water i.e. \[32:8\]. Now, use this ratio to calculate the amount of milk left in cask i.e. by subtracting the amount of milk replaced from the cask i.e. $\dfrac{4}{5} \times 8$ from 32. This step is repeated once again.

Complete step-by-step answer:
We are given 40 litre of milk in a cask and it is replaced with 8 litre water. This step is repeated two more times.
We have to find the number of litres of milk left in the cask.
First of all, when 8 litre is drawn out from 40 litre of milk and we left with $40 - 8$ litres of milk. Therefore, 32 litre of milk is left in the cask and when we add the same 8 litres of water. Thus, we are left with 32 litres of milk and 8 litres of water then the milk and water ratio will become \[32:8\]
When we reduce the ratio into the simplest form we get, \[4:1\].
Further, if we repeat this step,i.e. draw out 8 litres of mixture and replace it with water.
Now, this mixture also contains water then we have to find how much quantity of milk is replaced. We know the ratio of milk to water in the mixture before drawing 8 litres i.e. \[4:1\]. Here, there are 4 parts of milk out of 5 parts of mixture and 8 litre is taken from this mixture. Therefore, the amount of milk taken out is given as $ \Rightarrow \dfrac{4}{5} \times 8 = \dfrac{{32}}{5} = 6.4$ litres.
Therefore, we left with $32 - 6.4 = 25.6$ litres of milk left in the cask and the rest of mixture has water i.e. equal to $40 - 25.6 = 14.4$. Now, the ratio of milk to water is \[25.6:14.4\]
When we reduce the ratio into the simplest form we get, \[16:9\].

When we again repeat the above step, i.e. drawn 8 litres from the mixture and replace it with water.
Now, this mixture also contains water then we have to find how much quantity of milk is replaced. We know the ratio of milk to water in the mixture before drawing 8 litres i.e. \[16:9\]. Here, there are 16 parts of milk out of 25 parts of mixture and 8 litre is taken from this mixture. Therefore, the amount of milk taken out is given as $ \Rightarrow \dfrac{{16}}{{25}} \times 8 = \dfrac{{128}}{{25}} = 5.12$ litres.
Therefore, we finally left with $25.6 - 5.12 =20.48$ litres of milk left in the cask.
Hence, option C is the correct answer.

Note: Shortcut method:
This problem can be solved by a simpler method. We can see that every 8 litre is replaced with water from the 40 litres of mixture. Here, 8 litre is $5\% $ of 40 litres. Therefore, the quantity of milk in a 40 litre mixture is reduced by $5\% $ successively thrice to get the required answer.
Thus, the thought process could be as follow:
As, 10 percent of 40 is 4. Therefore, 20 percent of 40 is 8.
$ \Rightarrow 40 - 20\% \to 40 - 8 = 32$ litre
As, 10 percent of is 32 is 3.2. Therefore, 20 percent of 32 is 6.4.
$ \Rightarrow 32 - 20\% \to 32 - 6.4 = 25.6$ litre
As, 10 percent of 25.6 is 2.56. Therefore, 20 percent of 25.6 is 5.12.
$ \Rightarrow 25.6 - 20\% \to 25.6 - 5.12 = 20.48$ litre.