A and B can complete a work in 15 days and 24 days respectively. They start the work together and work for 4 days, after which both leave. Then, C joins and finishes the remaining work in 17 days. In how many days can C alone complete 50\% of the work?

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**Solution:**

1. Calculate the daily work rates of A and B:
   - A's rate = $\frac{1}{15}$ work/day
   - B's rate = $\frac{1}{24}$ work/day

2. Combined daily work of A and B:
   $$\frac{1}{15} + \frac{1}{24} = \frac{8}{120} + \frac{5}{120} = \frac{13}{120}$$

3. Work done by A and B in 4 days:
   $$4 	imes \frac{13}{120} = \frac{52}{120} = \frac{13}{30}$$

4. Remaining work after A and B leave:
   $$1 - \frac{13}{30} = \frac{17}{30}$$

5. C completes the remaining $\frac{17}{30}$ work in 17 days, so C's daily work rate is:
   $$\frac{17/30}{17} = \frac{1}{30}$$

6. To find the time C takes to complete 50\% (i.e., $\frac{1}{2}$) of the work:
   $$	ext{Time} = \frac{	ext{Work}}{	ext{Rate}} = \frac{1/2}{1/30} = 15 	ext{ days}$$

**Answer:** 15

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