Working alone, the times taken by Anu, Tanu, and Manu to complete any job are in the ratio 5:8:10. They accept a job which they can finish in 4 days if they all work together for 8 hours per day. However, Anu and Tanu work together for the first 6 days, working 6 hours 40 minutes per day. Then, the number of hours that Manu will take to complete the remaining job working alone is:

Question

Accepted Answer

To solve this problem:

1. **Work Efficiency Ratios**:  
   The times taken by Anu, Tanu, and Manu are in the ratio 5:8:10. Their efficiencies are inversely proportional to their times, so their efficiencies are in the ratio:  
   $$
   \frac{1}{5} : \frac{1}{8} : \frac{1}{10} = 8 : 5 : 4.
   $$

2. **Total Work Calculation**:  
   All three together can finish the job in 4 days working 8 hours per day. Hence, the total work (in terms of units) is:  
   $$
   	ext{Total Work} = (8 + 5 + 4) 	imes 8 	imes 4 = 17 	imes 32 = 544 	ext{ units}.
   $$

3. **Work Done by Anu and Tanu**:  
   Anu and Tanu work together for the first 6 days at 6 hours and 40 minutes per day (which is $$6 + \frac{40}{60} = \frac{20}{3}$$ hours per day). Their combined efficiency is:  
   $$
   (8 + 5) = 13 	ext{ units/hour}.
   $$  
   Work done by them in these 6 days is:  
   $$
   	ext{Work Done} = 13 	imes \frac{20}{3} 	imes 6 = 520 	ext{ units}.
   $$

4. **Remaining Work**:  
   The remaining work is:  
   $$
   	ext{Remaining Work} = 544 - 520 = 24 	ext{ units}.
   $$

5. **Manu's Time to Complete Remaining Work**:  
   Manu's efficiency is $$4$$ units/hour. Hence, the time taken by him to complete the remaining work is:  
   $$
   	ext{Time} = \frac{	ext{Remaining Work}}{	ext{Efficiency}} = \frac{24}{4} = 6 	ext{ hours}.
   $$

Thus, Manu will take **6 hours** to complete the remaining work.