Efficiency of Ritu is 60\% more than that of Rakesh. The difference between the number of days taken by them to complete a work is 18 days. If the efficiency of Rahul is 12.5\% less than that of Ritu, then find the number of days taken by all three working together to complete the work.

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**Solution:**

1. Let the efficiency of Rakesh be $5x$ units per day.

2. Since Ritu's efficiency is 60\% more than Rakesh's, her efficiency is:
   $$5x 	imes \frac{160}{100} = 8x 	ext{ units/day}$$

3. Rahul's efficiency is 12.5\% less than Ritu's, so:
   $$8x 	imes \frac{7}{8} = 7x 	ext{ units/day}$$

4. The ratio of time taken by Rakesh to Ritu to complete the work is inverse of their efficiencies:
   $$	ext{Time ratio} = \frac{8x}{5x} = \frac{8}{5}$$

5. Given the difference in days taken by Rakesh and Ritu is 18 days:
   $$8k - 5k = 18 \implies 3k = 18 \implies k = 6$$

6. Total work can be expressed as:
   $$	ext{Work} = 	ext{Time} 	imes 	ext{Efficiency} = 8k 	imes 5x = 8 	imes 6 	imes 5x = 240x 	ext{ units}$$

7. Combined efficiency of all three:
   $$5x + 8x + 7x = 20x 	ext{ units/day}$$

8. Time taken by all three together to complete the work:
   $$\frac{	ext{Total work}}{	ext{Combined efficiency}} = \frac{240x}{20x} = 12 	ext{ days}$$

**Answer:** 12 days

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