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

The correct answer is: 6
Working alone,the times taken by Anu,Tanu, and Manu to complete a job are in the ratio \(5:8:10\). When they work together for 8 hours per day,
they can finish a job in 4 days. However, a specific scenario is presented where Anu and Tanu work together for the first 6 days, each working 6
hours and 40 minutes per day. The goal is to determine the number of hours Manu will take to complete the remaining job working alone.
Let's break down the solution step by step:
1. Time Ratios:
\(Anu : Tanu : Manu = 5 : 8 : 10\)
2. Total Work:
Let the total work be W, and the times taken by Anu, Tanu, and Manu be 5x, 8x, and 10x, respectively.
W=LCM(5x, 8x, 10x)=40x
3. Individual Rates:
Anu's rate=8 units per hour
Tanu's rate=5 units per hour
Manu's rate=4 units per hour
4. Combined Rate:
Together,their combined rate is (8+5+4)=17 units per hour.
5. Time Calculation:
It is given that they can complete the job in 32 hours when working together:
Total work=\(Rate\times{Time}\)
\(40x = 17\times32\)
\(x=\frac{68}{5}=13.6\)
6. Anu and Tanu's Work:
Anu and Tanu work together for 6 days, each working 6 hours and 40 minutes per day:
Work done by Anu and Tanu together=\((8+5)units\times40 hours=520 units\)
7. Remaining Work:
Remaining work=Total work-Work done by Anu and Tanu
Remaining work=40x-520 units
8. Time Taken by Manu:
Manu works alone, and his rate is 4 units per hour.
Remaining work=Manu's rate * Time taken by Manu
40x-520=4y
\(y=\frac{(40x - 520)}{4}\)
y=10x-130
9. Substituting x:
\(y=10\times13.6 - 130 = 136 - 130 = 6 hours\)
Therefore, Manu will take 6 hours to complete the remaining job working alone.

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 \(= 4\times 8\times(8+5+4)\)
\(= 4\times 8\times17\)
\(= 544\)
Work done by Anu and Tan
\(=  6×6.67×(8+5)\)
\(=  6×6.67×13\)
\(= 520\)
Remaining work,
\(= 544 – 520\)
\(= 24\)
Time taken by Manu to complete the work,
\(= \frac {24}{4}\)
\(= 6\) hours