D01-29

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?

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

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