A and B can complete a work in 15 days and 10 days, respectively. They started doing the work together but after 4 days, B had to leave, and A completed the remaining work alone. The entire work was completed in ______________ days.

The correct option is 9.

Let us solve the problem step-by-step to understand why this is the correct answer.

First, let the total work be represented as a common multiple of the days taken by A and B individually.
Time taken by A to complete the work alone = 15 days
Time taken by B to complete the work alone = 10 days
Let us assume the total work is the Least Common Multiple (LCM) of 15 and 10, which is 30 units.

Now, we calculate the efficiency (work done per day) of A and B:

Efficiency of A = $$\frac{30}{15}=2$$ units/day.

Efficiency of B = $$\frac{30}{10}=3$$ units/day.

Together, A and B can do:

Combined efficiency of (A + B) = 2 + 3 = 5 units/day.

They worked together for 4 days before B left.
Work completed in these 4 days by A and B together:

Work done = 4 days × 5 units/day = 20 units.

Remaining work after 4 days:

Remaining work = 30 units - 20 units = 10 units.

This remaining work of 10 units was completed by A alone.
Time taken by A to complete the remaining work:

Time taken by A = $$\frac{\text{Remaining work}}{\text{Efficiency of A}}=\frac{10}{2}=5$$ days.

To find the total number of days taken to complete the entire work, we add the time they worked together to the time A worked alone:

Total days = 4 days (together) + 5 days (A alone) = 9 days.

Therefore, the entire work was completed in 9 days.