Rahul and Rajeev together can finish a work in 17 days, while Rahul, Rajeev, and Raj together can complete the same work in 12 days. If Rahul is 60\% more efficient than Raj, find the number of days Rajeev alone will take to complete the entire work.

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**Solution:**

1. Let the efficiencies of Rahul, Rajeev, and Raj be $R_h$, $R_v$, and $R_j$ respectively.

2. Given Rahul is 60\% more efficient than Raj:
   $$R_h = R_j + 0.6 R_j = 1.6 R_j$$

3. Rahul and Rajeev together can complete the work in 17 days, so their combined efficiency is:
   $$\frac{1}{17} = R_h + R_v$$

4. Rahul, Rajeev, and Raj together can complete the work in 12 days, so:
   $$\frac{1}{12} = R_h + R_v + R_j$$

5. Substitute $R_h = 1.6 R_j$ into the equations:
   - From step 3:
     $$\frac{1}{17} = 1.6 R_j + R_v$$
   - From step 4:
     $$\frac{1}{12} = 1.6 R_j + R_v + R_j = 2.6 R_j + R_v$$

6. Subtract the first equation from the second:
   $$\frac{1}{12} - \frac{1}{17} = (2.6 R_j + R_v) - (1.6 R_j + R_v) = 1.0 R_j$$

7. Calculate the left side:
   $$\frac{1}{12} - \frac{1}{17} = \frac{17 - 12}{12 	imes 17} = \frac{5}{204}$$

So,
   $$R_j = \frac{5}{204}$$

8. Using $R_h = 1.6 R_j = 1.6 	imes \frac{5}{204} = \frac{8}{204} = \frac{2}{51}$

9. From step 3:
   $$\frac{1}{17} = R_h + R_v = \frac{2}{51} + R_v$$

Therefore,
   $$R_v = \frac{1}{17} - \frac{2}{51} = \frac{3}{51} - \frac{2}{51} = \frac{1}{51}$$

10. Efficiency of Rajeev is $\frac{1}{51}$, so time taken by Rajeev alone to complete the work is:
    $$	ext{Time} = \frac{1}{R_v} = 51 	ext{ days}$$

**Answer:** 51 days

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