Sam can complete a job in 20 days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna is the same job. The undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayna on the second day, Mohit and Ayna on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is
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Official Correct Answer: B. Okay, so I'm trying to solve this CAT question about work rates. Sam can finish a job in 20 days alone. Mohit is twice as fast as Sam, and thrice as fast as Ayna. They work in a pattern: Sam and Mohit on day one, Sam and Ayna on day two, Mohit and Ayna on day three, and this cycle repeats until the job is done. We find the fraction of the work done by Sam.
First, I need to figure out the work rates of each person. Work rate is usually calculated as 1 divided by the time they take to finish the job alone. So, Sam's rate is 1/20 per day.
Mohit is twice as fast as Sam. If Sam takes 20 days, Mohit would take half the time, which is 10 days. So, Mohit's rate is 1/10 per day.
Now, Mohit is thrice as fast as Ayna. That means Ayna is one-third as fast as Mohit. Since Mohit takes 10 days, Ayna would take 3 times that, which is 30 days. So, Ayna's rate is 1/30 per day.
We have their rates:
- Sam: 1/20 per day
- Mohit: 1/10 per day
- Ayna: 1/30 per day
Next, the work pattern is three days: days 1, 2, 3 are Sam+Mohit, Sam+Ayna, Mohit+Ayna, and then it repeats.
We calculate how much work is done each day and then figure out how many such cycles are needed to complete the job. Then, I can find out how much work Sam did in total.
Compute the work done each day:
Day 1: Sam and Mohit working together. Their combined rate is 1/20 + 1/10. Compute that:
1/20 + 1/10 = 1/20 + 2/20 = 3/20. So, they complete 3/20 of the work on day 1.
Day 2: Sam and Ayna. Their combined rate is 1/20 + 1/30. Let me find a common denominator, which is 60.
1/20 = 3/60 and 1/30 = 2/60. So, 3/60 + 2/60 = 5/60 = 1/12. So, they do 1/12 on day 2.
Day 3: Mohit and Ayna. Their combined rate is 1/10 + 1/30. Again, common denominator is 30.
1/10 = 3/30 and 1/30 is 1/30. So, 3/30 + 1/30 = 4/30 = 2/15. So, they do 2/15 on day 3.
Now, let me find the total work done in three days. That would be 3/20 + 1/12 + 2/15.
To add these fractions, I need a common denominator. , denominators are 20, 12, 15. The least common multiple is 60.
Convert each fraction:
3/20 = 9/60
1/12 = 5/60
2/15 = 8/60
Adding them up: 9 + 5 + 8 = 22. So, 22/60 = 11/30.
So, in three days, they complete 11/30 of the work.
Now, the total work is 1. So, how many full cycles of three days are needed?
Let the number of full cycles as n. So, n * 11/30 ≤ 1.
We can solve for n:
n ≤ 1 / (11/30) = 30/11 ≈ 2.727.
So, n can be 2 full cycles, which would complete 2 * 11/30 = 22/30 = 11/15 of the work.
Then, the remaining work is 1 - 11/15 = 4/15.
Now, after 2 full cycles, the next day is day 1 of the next cycle, which is Sam and Mohit working together.
Their combined rate is 3/20 per day. So, how much time do they need to finish 4/15 of the work?
Let the time as t days.
So, 3/20 * t = 4/15
Solving for t:
t = (4/15) / (3/20) = (4/15) * (20/3) = (80/45) = 16/9 days ≈ 1.777 days.
But since they can't work a fraction of a day in this context, we need to see how much work is done on the third day.
Wait, actually, in the problem, the work is done in cycles of three days, but if the remaining work is less than a day's work, they might not need a full day.
But let me think again. After 2 cycles, 11/15 is done. Remaining is 4/15.
On day 7 (which is the start of the 3rd cycle), Sam and Mohit work together. Their rate is 3/20 per day. So, how much can they do in a day? 3/20.
But 3/20 is equal to 9/60, and 4/15 is equal to 16/60. So, 9/60 is less than 16/60. So, they can't finish the remaining work in one day. Hmm, that seems conflicting.
Wait, maybe I made a mistake in calculating the remaining work.
Wait, 1 - 11/15 = 4/15. 4/15 is approximately 0.2667.
Sam and Mohit together do 3/20 per day, which is 0.15 per day. So, 0.2667 / 0.15 ≈ 1.777 days. So, they need a little over 1.777 days to finish the remaining work.
But since the work is done in full days, perhaps they work for two more days, but that would exceed the work. Hmm, this is getting confusing.
Wait, maybe I should model it differently. Let's think about the total work done after each