Explanations:In work and time problems, the work rate of each individual is calculated as the reciprocal of the time they take to complete the work alone. For example, if A can complete a piece of work in 12 days, A's work rate is 1/12 of the work per day. Similarly, if B can complete the same piece of work in 24 days, B's work rate is 1/24 of the work per day.
When A and B work together, their combined work rate is the sum of their individual work rates. Therefore, the combined work rate of A and B is:
\[
\text{Combined work rate} = \frac{1}{12} + \frac{1}{24} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8}
\]
This means that together, A and B can complete 1/8 of the work per day. If they work together for 4 days, the amount of work they complete is:
\[
\text{Work done in 4 days} = 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2}
\]
So, A and B together complete half of the work in 4 days. The remaining work is:
\[
\text{Remaining work} = 1 - \frac{1}{2} = \frac{1}{2}
\]
After A leaves, B continues to work alone. B's work rate is 1/24 of the work per day. To find out how many days B will take to complete the remaining half of the work, we use the formula:
\[
\text{Time taken by B} = \frac{\text{Remaining work}}{\text{B's work rate}} = \frac{\frac{1}{2}}{\frac{1}{24}} = \frac{1}{2} \times 24 = 12 \text{ days}
\]
Therefore, B will take 12 days to complete the remaining half of the work alone. This confirms that the correct answer is 12 days.