A completes $${5 \over 6}$$ th of a given task in 10 days and is then replaced by B. The entire task is completed in 13 days. What is the respective ratio of the number of days in which A and B independently can complete the entire task?

Let total work to be done = 6 units

=> A completes $$\frac{5}{6} \times 6 = 5$$ units in 10 days

=> A's efficiency = $$\frac{5}{10} = \frac{1}{2}$$ units /day

Now, B finishes $$\frac{1}{6}$$th of the task in 3 days

=> B completes $$\frac{1}{6} \times 6 = 1$$ units in 3 days

=> B's efficiency = $$\frac{1}{3}$$ units /day

Now, time taken by A alone to complete the entire task = $$\frac{6}{\frac{1}{2}} = 12$$ days

Time taken by B alone to complete the entire task = $$\frac{6}{\frac{1}{3}} = 18$$ days

$$\therefore$$ Required ratio = $$\frac{12}{18} = 2 : 3$$