A and B can complete a work in 15 days and 10 days, respectively. They started doing the work together but after 2 days, B had to leave, and A completed the remaining work alone. The whole work was completed in ______________ days.

The correct option is 12.

Let's break down the solution step-by-step to understand why the whole work was completed in 12 days.

Step 1: Determine the rate of work for A and B.
Let the total work to be completed be represented by 1 unit.
Since A can complete the entire work in 15 days, the work done by A in 1 day is:
$$W_A=\frac{1}{15}$$ Similarly, since B can complete the entire work in 10 days, the work done by B in 1 day is:
$$W_B=\frac{1}{10}$$

Step 2: Calculate the work done together in the first 2 days.
When working together, the amount of work completed by A and B in 1 day is:
$$W_{A+B}=\frac{1}{15}+\frac{1}{10}$$ To add these fractions, we find a common denominator, which is 30:
$$W_{A+B}=\frac{2+3}{30}=\frac{5}{30}=\frac{1}{6}$$ In 2 days, the work completed by both of them together is:
$$\text{Work done in 2 days}=2\times \frac{1}{6}=\frac{1}{3}$$

Step 3: Calculate the remaining work.
After 2 days, B leaves. The remaining work to be completed is:
$$\text{Remaining work}=1-\frac{1}{3}=\frac{2}{3}$$

Step 4: Determine the time taken by A to finish the remaining work alone.
Since A completes 1/15 of the work in one day, the number of days A needs to complete the remaining 2/3 of the work is:
$$\text{Days taken by A}=\frac{\text{Remaining work}}{W_A}=\frac{2/3}{1/15}=\frac{2}{3}\times 15=2\times 5=10\text{days}$$

Step 5: Calculate the total time taken to complete the entire work.
The total time consists of the initial 2 days when A and B worked together, plus the 10 days A worked alone to finish the rest:
$$\text{Total days}=2+10=12\text{days}$$