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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

Updated On: Jan 15, 2026
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Solution and Explanation

Given:

  • The ratio of time taken by Anu, Tanu, and Manu to complete a job individually is \(5 : 8 : 10\).
  • When working together, they work 8 hours per day and complete the job in 4 days.
  • In a specific scenario, Anu and Tanu work for 6 days, 6 hours and 40 minutes each day.
  • The objective is to determine the time Manu requires to complete the remaining work alone.

Step 1: Time Ratios

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

Step 2: Total Work Calculation

The total work is considered as the Least Common Multiple (LCM) of their individual times: \(\text{LCM}(5x, 8x, 10x) = 40x\) units.

Step 3: Individual Work Rates (units/hour)

  • Anu's rate: \(\frac{40x}{5x} = 8\) units/hour
  • Tanu's rate: \(\frac{40x}{8x} = 5\) units/hour
  • Manu's rate: \(\frac{40x}{10x} = 4\) units/hour

Step 4: Combined Work Rate

When working together, their combined rate is \(8 + 5 + 4 = 17\) units/hour.

Step 5: Total Work Done Together

They worked for \(8 \text{ hours/day} \times 4 \text{ days} = 32\) hours.

The total work completed is \(17 \text{ units/hour} \times 32 \text{ hours} = 544\) units.

Since the total work is \(40x\), we have: \(40x = 544\), which gives \(x = \frac{544}{40} = \frac{68}{5} = 13.6\).

Step 6: Work Done by Anu and Tanu

They worked for 6 days, at a rate of 6 hours and 40 minutes per day.

Convert 6 hours 40 minutes to hours: \(6 + \frac{40}{60} = 6 + \frac{2}{3} = \frac{20}{3}\) hours/day.

Total hours worked by Anu and Tanu: \(6 \text{ days} \times \frac{20}{3} \text{ hours/day} = 40\) hours.

Their combined rate is \(8 + 5 = 13\) units/hour.

Work done by Anu and Tanu: \(13 \text{ units/hour} \times 40 \text{ hours} = 520\) units.

Step 7: Remaining Work

Total work = 544 units.

Remaining work = \(544 \text{ units} - 520 \text{ units} = 24\) units.

Step 8: Manu's Time to Complete Remaining Work

Manu's work rate is 4 units/hour.

Time required for Manu to complete the remaining work = \(\frac{24 \text{ units}}{4 \text{ units/hour}} = 6\) hours.

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

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