A, B, and C together can complete $66 \frac{2}{3}\%$ of the work in 2 days, and A and B together can complete 30\% of the work in 3 days. Find the time taken by C alone to complete $4 \frac{2}{3}$ of the total work.

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**Solution:**

1. Given that A, B, and C together complete $66 \frac{2}{3}\%$ of the work in 2 days. Convert $66 \frac{2}{3}\%$ to fraction:
   $$66 \frac{2}{3}\% = \frac{2}{3}$$

2. Work done by A, B, and C in 2 days is $\frac{2}{3}$ of total work. Therefore, total work will be completed by them in:
   $$	ext{Total time} = \frac{2}{\frac{2}{3}} = 3 	ext{ days}$$

3. A and B together complete 30\% of the work in 3 days. Convert 30\% to fraction:
   $$30\% = \frac{30}{100} = \frac{3}{10}$$

4. Time taken by A and B together to complete the whole work:
   $$	ext{Total time} = \frac{3}{\frac{3}{10}} = 10 	ext{ days}$$

5. Let total work be 30 units (LCM of 3 and 10 for convenience).
   - Efficiency of A, B, and C together:
     $$= \frac{30}{3} = 10 	ext{ units/day}$$
   - Efficiency of A and B together:
     $$= \frac{30}{10} = 3 	ext{ units/day}$$

6. Efficiency of C alone:
   $$= 10 - 3 = 7 	ext{ units/day}$$

7. The required time for C alone to complete $4 \frac{2}{3}$ of the work:
   Convert $4 \frac{2}{3}$ to improper fraction:
   $$4 \frac{2}{3} = \frac{14}{3}$$

Total work units for $4 \frac{2}{3}$ work:
   $$= \frac{14}{3} 	imes 30 = 140 	ext{ units}$$

8. Time taken by C alone:
   $$= \frac{140}{7} = 20 	ext{ days}$$

**Answer:** 20 days

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