X is four times as efficient as Y in respect of doing a particular work. Working together they complete the work in 16 days. In how many days y working alone will be able to half the work?

Step 1: Understand the problem

The problem states that X is four times as efficient as Y. This means that if Y does 1 unit of work, X does 4 units of work.

Step 2: Calculate the total work done

When X and Y work together, they complete the work in 16 days. This means that in one day, they do 1/16 of the work.

Step 3: Calculate the work done by Y in one day

Since X is four times as efficient as Y, the work done by X and Y together in one day is 5 units (4 units by X and 1 unit by Y). So, 1/16 of the total work is equal to 5 units.

Step 4: Calculate the work done by Y alone in one day

If 5 units of work is equal to 1/16 of the total work, then 1 unit of work (which is what Y does in one day) is equal to 1/16 divided by 5, which is 1/80 of the total work.

Step 5: Calculate the time taken by Y to do half the work

If Y does 1/80 of the work in one day, then to do half the work (which is 1/2), Y will take 80 times 1/2, which is 40 days.

So, Y alone will be able to do half the work in 40 days.