Question

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

Answer 1

Given:

• Time ratios for Anu, Tanu, and Manu to complete a job individually: \(5 : 8 : 10\)
• They work together for 8 hours/day and complete the job in 4 days.
• In a specific case, Anu and Tanu work for 6 days, 6 hours 40 minutes each day.
• We need to find the number of hours Manu will take to complete the remaining work alone.

Step 1: Time Ratios

Let the actual time taken by Anu, Tanu, and Manu be \(5x, 8x, 10x\) respectively.

Step 2: Total Work

Take total work as LCM of the times: \(\text{LCM}(5x, 8x, 10x) = 40x\)

Step 3: Individual Work Rates (units/hour)

• Anu: \(\frac{40x}{5x} = 8\)
• Tanu: \(\frac{40x}{8x} = 5\)
• Manu: \(\frac{40x}{10x} = 4\)

Step 4: Total Rate when All Work Together

Combined rate = \(8 + 5 + 4 = 17\) units/hour

Step 5: Total Work Done Together

They work 8 hours per day for 4 days: \(8 \times 4 = 32\) hours
Total work = \(17 \times 32 = 544\) units
But total work = \(40x\), so: 

\[40x = 544 \Rightarrow x = \frac{544}{40} = \frac{68}{5} = 13.6\]

Step 6: Anu and Tanu’s Work

They work together for 6 days, 6 hours 40 minutes per day.
Convert 6 hours 40 minutes to hours: \(6 + \frac{40}{60} = \frac{20}{3}\) hours/day
Total hours worked = \(6 \times \frac{20}{3} = 40\) hours
Combined rate = \(8 + 5 = 13\) units/hour
Work done = \(13 \times 40 = 520\) units

Step 7: Remaining Work

Total work = \(40x = 544\)
Remaining work = \(544 - 520 = 24\) units

Step 8: Manu’s Time to Finish Remaining Work

Manu’s rate = 4 units/hour
Time = \(\frac{24}{4} = 6\) hours

Final Answer: \(\boxed{6 \text{ hours}}\)

Answer 2

Given that, the times taken by Anu, Tanu, and Manu to complete any job are in the ratio \(5:8:10\)
Therefore, the efficiency will be in the ratio,  
\(= \frac{40}{5} : \frac{40}{8} : \frac{40}{10}\)
\(= 8 : 5 : 4\)
Hence, Anu does \(8\) units per hour, Tanu does \(5\) units per hour, and Manu does \(4\) units per hour.

Total work \(= 4 \times 8 \times (8 + 5 + 4)\)
\(= 4 \times 8 \times 17 = 544\)

Work done by Anu and Tanu in \(6 \times 6.67\) hours:
\(= 6 \times 6.67 \times (8 + 5)\)
\(= 6 \times 6.67 \times 13 = 520\)

Remaining work:
\(= 544 - 520 = 24\)
Time taken by Manu to complete remaining work:
\(= \frac{24}{4} = 6\) hours