Explanation: To solve this problem, we need to understand the concept of work and time, specifically how to calculate the combined work rate and the remaining work after a certain period.
First, let's determine the individual work rates of A and B. A can complete the work in 12 days, so A's work rate is \( \frac{1}{12} \) of the work per day. Similarly, B can complete the work in 24 days, so B's work rate is \( \frac{1}{24} \) of the work per day.
When A and B work together, their combined work rate is the sum of their individual work rates:
\[ \text{Combined work rate} = \frac{1}{12} + \frac{1}{24} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8} \]
This means that together, A and B can complete \( \frac{1}{8} \) of the work per day.
Next, we calculate the amount of work completed by A and B together in 4 days:
\[ \text{Work completed in 4 days} = 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]
So, after 4 days, half of the work is completed, and the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{2} = \frac{1}{2} \]
Now, B will complete the remaining \( \frac{1}{2} \) of the work alone. Since B's work rate is \( \frac{1}{24} \) of the work per day, the time B needs to complete the remaining work is:
\[ \text{Time for B to complete remaining work} = \frac{\frac{1}{2}}{\frac{1}{24}} = \frac{1}{2} \times 24 = 12 \text{ days} \]
Therefore, the correct answer is that B will complete the remaining work in 12 days. This matches option A.
In summary, the key steps are:
1. Calculate the individual work rates of A and B.
2. Determine the combined work rate of A and B.
3. Calculate the work completed by A and B together in 4 days.
4. Find the remaining work after 4 days.
5. Calculate the time B needs to complete the remaining work alone.
This problem tests the understanding of work and time concepts, including combined work rates and the calculation of remaining work.