A alone can do a work in 24 days. Time taken by A in completing $ \frac{1}{3} $ of the work is equal to the time taken by B in completing $ \frac{1}{2} $ of the work. In what time will A and B together complete the work?

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**Solution:**

1. Given, A alone can complete the work in 24 days.

2. Time taken by A to complete $ \frac{1}{3} $ of the work:
   $$
   24 \times \frac{1}{3} = 8 \text{ days}
   $$

3. This time (8 days) is equal to the time taken by B to complete $ \frac{1}{2} $ of the work.

4. Let the total time taken by B to complete the whole work be $ x $ days.

Since B completes $ \frac{1}{2} $ work in 8 days,
   $$
   \frac{1}{2} \text{ work} = 8 \text{ days} \implies x = 8 \times 2 = 16 \text{ days}
   $$

5. Work rates:
   - A's rate = $ \frac{1}{24} $ work/day
   - B's rate = $ \frac{1}{16} $ work/day

6. Combined rate of A and B:
   $$
   \frac{1}{24} + \frac{1}{16} = \frac{2}{48} + \frac{3}{48} = \frac{5}{48} \text{ work/day}
   $$

7. Time taken by A and B together to complete the work:
   $$
   \frac{1}{\frac{5}{48}} = \frac{48}{5} = 9.6 \text{ days}
   $$

**Answer:** $ \frac{48}{5} $ days

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