X alone can complete a piece of work in 16 days, while Y alone can complete the same work in 24 days. They work on alternate days starting with Y. In how many days will 50% of the total work be completed?

Let's assume the total work is 48 units.

Y alone can complete the same work in 24 days.

Efficiency of Y = $$\frac{48}{24}$$ =  2 units/day

X alone can complete a piece of work in 16 days.

Efficiency of X = $$\frac{48}{16}$$ = 3 units/day

They work on alternate days starting with Y to do 50% of the total work.

50% of the total work = 50% of 48 = $$\frac{50}{100}\times48$$ = 24 units

Work done by Y on first day = 2 units

Work done by X on second day = 3 units

So the work done in first two days = 2+3 = 5 units

work done in first eight days = $$5\times4$$ = 20 units.

Now on ninth day Y will come and do 2 units of work.

After that (24-20-2) = 2 units of work will be remaining which will be done by X in $$\frac{2}{3}$$ days.

Hence the time taken to do 50% of the total work on an alternative basis = $$8+1+\frac{2}{3}$$

= $$9+\frac{2}{3}$$

= $$\frac{27+2}{3}$$

= $$\frac{29}{3}$$ days