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

Rahul, Rakshita and Gurmeet, working together, would have taken more than 7 days to finish a job. On the other hand, Rahul and Gurmeet, working together would have taken less than 15 days to finish the job. However, they all worked together for 6 days, followed by Rakshita, who worked alone for 3 more days to finish the job. If Rakshita had worked alone on the job then the number of days she would have taken to finish the job, cannot be

Answer 1

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.

Solution and Explanation