X alone can complete a work in 18 days and Y alone can do the same work in half time of the time taken by X. Working together what part of the same work they can complete in a day ?

1/6

time ∝ $\frac{1}{efficiency}$

               X  :  Y

Time       18  :  9   = 2  :  1

Efficiency  1  :  2     

Let total work = 18

Efficiency of X + Y = 3

Total time taken = $\frac{total\;work}{efficiency}$

                        = $\frac{18}{3}$ = 6

Work done in 1 day = $\frac{1}{total\;time\;taken}$

Thus, work done in 1 day = $\frac{1}{6}$