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I think this is a high-quality question and I agree with explanation.
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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I think this is a high-quality question and I agree with explanation.
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I think this is a high-quality question and I agree with explanation.
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I think this is a high-quality question and I agree with explanation.
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I think this is a high-quality question and I agree with explanation.
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Bunuel: If we know that T/M+T/J + R/J=1 and we also know that both did the same job, can't we equate T/M=T/J + R/J? If yes, couldn't we just solve for R/J=T/M-T/J and then use it in our initial equation providing T/M+T/J + T/M-T/J=1 leading to 2T/M=1 so that either information about T or M would be sufficient to solve for all variables? Then S1 would be sufficient. Where did I go wrong?
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Bunuel
Official Solution:


Mac can complete a job in \(M\) days and Jack can complete the same job in \(J\) days. After working together for \(T\) days, Mac left and Jack worked alone to complete the remaining work in \(R\) days. If Mac and Jack completed an equal amount of work, how many days would it take Jack to complete the entire job working alone?

Mac can complete the job in \(M\) days, which means his rate is \(\frac{1}{M}\) job/day.;

Jack can complete the job in \(J\) days, which means his rate is \(\frac{1}{J}\) job/day.

After working together for \(T\) days, Mac left and Jack worked alone to complete the remaining work in \(R\) days. Therefore, Mac worked for \(T\) days only and completed \(\frac{T}{M}\) part of the job, while Jack worked for \(T+R\) days and completed \(\frac{T+R}{J}\) part of the job.

Since Mac and Jack completed an equal amount of work (which means each of them did half of the job), then \(\frac{T}{M}=\frac{1}{2}\) and \(\frac{T+R}{J}=\frac{1}{2}\). Solving for \(T\) gives \(T=\frac{M}{2}\) and \(T=\frac{J}{2}-R\). Therefore, \(\frac{M}{2}=\frac{J}{2}-R\), which simplifies to \(J=M+2R\).

(1) \(M = 20\) days. Not sufficient, we still need the value of \(R\).

(2) \(R = 10\) days. Not sufficient, we still need the value of \(M\).

(1)+(2) Using the equation \(J=M+2R\) and the values \(M=20\) and \(R=10\), we can solve for \(J\) to get \(J=40\). Therefore, Jack would need \(J=40\) days to complete the entire job working alone. Sufficient.


Answer: C
Bunuel: If we know that T/M+T/J + R/J=1 and we also know that both did the same job, can't we equate T/M=T/J + R/J? If yes, couldn't we just solve for R/J=T/M-T/J and then use it in our initial equation providing T/M+T/J + T/M-T/J=1 leading to 2T/M=1 so that either information about T or M would be sufficient to solve for all variables? Then S1 would be sufficient. Where did I go wrong?

Yes, as shown in the solution, we can get \(T = \frac{M}{2}\) (and \(T = \frac{J}{2} - R\)) from the stem. However, from (1), we would still only know M and T, while J and R would remain unknown. Notice that 10/20 + 10/J + R/J = 1 and 10 = J/2 - R are essentially the same equations (simplifying 10/20 + 10/J + R/J = 1 gives 10 = J/2 - R), so you cannot solve for J and R.

Hope it's clear.
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I like the solution - it’s helpful.
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I did not quite understand the solution. Hi, I generally do this by the numerator method, so I assumed the work as MJ
Mac-- M time, J rate
Jack-- J time, M rate

So, MJ= (M+J)T + JR (both's rate of work into time and jack's individual work after mac left)
they both have done the same amount of work, hence
= MJ=2(T+R)J & MJ=2MT
= M=2(T+R) & J=2T

after this I am getting confused
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Vineet1102
I did not quite understand the solution. Hi, I generally do this by the numerator method, so I assumed the work as MJ
Mac-- M time, J rate
Jack-- J time, M rate

So, MJ= (M+J)T + JR (both's rate of work into time and jack's individual work after mac left)
they both have done the same amount of work, hence
= MJ=2(T+R)J & MJ=2MT
= M=2(T+R) & J=2T

after this I am getting confused

Can you please tell me what exactly is unclear in the solution ? Have you read the discission in the thread?
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I like the solution - it’s helpful.
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This is a great question that’s helpful for learning.
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Thanks, this explanation really helped. I was stuck with an extra variable while solving this.
Bunuel
Official Solution:


Mac can complete a job in \(M\) days and Jack can complete the same job in \(J\) days. After working together for \(T\) days, Mac left and Jack worked alone to complete the remaining work in \(R\) days. If Mac and Jack completed an equal amount of work, how many days would it take Jack to complete the entire job working alone?

Mac can complete the job in \(M\) days, which means his rate is \(\frac{1}{M}\) job/day.;

Jack can complete the job in \(J\) days, which means his rate is \(\frac{1}{J}\) job/day.

After working together for \(T\) days, Mac left and Jack worked alone to complete the remaining work in \(R\) days. Therefore, Mac worked for \(T\) days only and completed \(\frac{T}{M}\) part of the job, while Jack worked for \(T+R\) days and completed \(\frac{T+R}{J}\) part of the job.

Since Mac and Jack completed an equal amount of work (which means each of them did half of the job), then \(\frac{T}{M}=\frac{1}{2}\) and \(\frac{T+R}{J}=\frac{1}{2}\). Solving for \(T\) gives \(T=\frac{M}{2}\) and \(T=\frac{J}{2}-R\). Therefore, \(\frac{M}{2}=\frac{J}{2}-R\), which simplifies to \(J=M+2R\).

(1) \(M = 20\) days. Not sufficient, we still need the value of \(R\).

(2) \(R = 10\) days. Not sufficient, we still need the value of \(M\).

(1)+(2) Using the equation \(J=M+2R\) and the values \(M=20\) and \(R=10\), we can solve for \(J\) to get \(J=40\). Therefore, Jack would need \(J=40\) days to complete the entire job working alone. Sufficient.


Answer: C
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