The problem says:
Amal and Vimal together can complete a task in 150 days, while Vimal and Sunil together can complete the same task in 100 days. Then, there's a scenario where Amal works for 75 days, Vimal takes over for 135 days, and finally, Sunil completes the remaining task in 45 days.
Now, the question is asking: If instead, Amal had started the task alone and worked on all days, Vimal had worked on every second day, and Sunil had worked on every third day, how many days would it take to complete the task?
The correct answer is given as 139, but I need to figure out how to get there.
First, I think We find the individual work rates of Amal, Vimal, and Sunil. Since they're working together in pairs, I can set up equations based on their combined work rates.
Let me denote:
Amal's work rate as A (tasks per day)
From the problem:
1
Amal and Vimal together can complete the task in 150 days. So, their combined work rate is 1/150 tasks per day.
Therefore, A + V = 1/150.
2
Vimal and Sunil together can complete the task in 100 days. So, their combined work rate is 1/100 tasks per day.
Therefore, V + S = 1/100.
Now, I have two equations:
But I have three variables here, so I need another equation. Let's look at the second scenario given in the problem.
In the second scenario:
Vimal works for 135 days.
So, the total work done is the sum of each person's contribution.
Total work = 75A + 135V + 45*S = 1 task.
So, equation 3 is: 75A + 135V + 45S = 1.
Now, I have three equations:
I need to solve these equations to find A, V, and S.
Let me try to express A and S in terms of V.
From equation 1: A = 1/150 - V.
From equation 2: S = 1/100 - V.
Now, substitute A and S into equation 3.
75(1/150 - V) + 135V + 45(1/100 - V) = 1.
Compute each term step by step.
First, compute 75(1/150 - V): 75(1/150) = 75/150 = 0.5 75*(-V) = -75V
So, 75*(1/150 - V) = 0.5 - 75V.
Next, compute 135V: That's just 135V.
Then, compute 45(1/100 - V): 45(1/100) = 45/100 = 0.45 45*(-V) = -45V
So, 45*(1/100 - V) = 0.45 - 45V.
Now, add all these together: (0.5 - 75V) + 135V + (0.45 - 45V) = 1.
Combine like terms: 0.5 + 0.45 = 0.95 -75V + 135V -45V = 15V
So, the equation becomes: 0.95 + 15V = 1.
Subtract 0.95 from both sides: 15V = 1 - 0.95 = 0.05.
Therefore, V = 0.05 / 15 = 0.003333... which is 1/300.
So, Vimal's work rate is 1/300 tasks per day.
Now, substitute V back into equation 1 to find A: A = 1/150 - V = 1/150 - 1/300.
Convert to common denominator: 1/150 = 2/300, so 2/300 - 1/300 = 1/300.
So, Amal's work rate is 1/300 tasks per day.
Similarly, substitute V into equation 2 to find S: S = 1/100 - V = 1/100 - 1/300.
Convert to common denominator: 1/100 = 3/300, so 3/300 - 1/300 = 2/300 = 1/150.
So, Sunil's work rate is 1/150 tasks per day.
Now, we have: A = 1/300 V = 1/300 S = 1/150
Now, the problem asks: If Amal had started the task alone and worked on all days, Vimal had worked on every second day, and Sunil had worked on every third day, then the number of days required to complete the task would have been?
So, in this new scenario, each day:
Vimal works every second day (i.e., day 2, 4, 6, etc.).
Sunil works every third day (i.e., day 3, 6, 9, etc.).
We need to find the total number of days, say D, required to complete 1 task under this schedule.
Let me think about how much each person contributes in D days.
Amal works every day, so his total contribution is DA = D(1/300).
Vimal works every second day, so he works on floor(D/2) days. His contribution is floor(D/2)V = floor(D/2)(1/300).
Similarly, Sunil works every third
