Data Sufficiency

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

(1) M = 20 days
(2) R = 10 days

A. Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
B. Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
D. EACH statement ALONE is sufficient
E. Statements (1) and (2) TOGETHER are NOT sufficient

The setup for these problems is output/time:

1/M + 1/J = 1/(Total Time)

1) Insufficient: You have no idea Jack's speed nor can you calculate it without The total time or R
2) Insufficient: The speed of both Jack and Mac can vary with this information

Together, you know that Mac can do the job in 20 days and that Jack takes 10 days to finish the remaining work. However, you do not know the amount of the remaining work, because you do not know how long Mac and Jack worked together, which would tell you their respective rates.

BTW, the answer is thus E.

Mac can finish the job in M days. So, in one day Mac can complete 1/M th of the Job.
Jack can finish the job in J days. So, in one day Jack can complete 1/J th of the Job.
Mac + Jack can complete 1/M + 1/J th of the Job in a day.
After working together for T days, Mac + Jack can complete T*(1/M + 1/J)th of the work.
Jack alone completed R/J th of the job.

So, T*(1/M + 1/J) + R/J = 1
Total work done by Jack = T/J + R/J
Total work done by Mac = T/M
Total work done by Jack = Total work done by Mac
T/J + R/J = T/M

Now, the question can be rephrased to, If T/M + T/J + R/J = 1 and T/J + R/J = T/M, then what is the value of J?'

(1)M = 20 days

T/M + T/J + R/J = 1
T/J + R/J = T/M
So, T/M + T/M = 1
2T = M = 20. T = 10.
We know T/J + R/J = T/M, 10/J + R/J = 10/20. Since we don't know the value of R, statement I is insufficient to answer the question.

(2)R = 10 days

T/M + T/J + R/J = 1
T/J + R/J = T/M
So, 2*(T/J + R/J) = 1
T/J + R/J = 1/2
T/J + 10/J = 1/2. Since we don't know the value of T, statement II is insufficient to answer the question.

From I and II

T/J + 10/J = 1/2 and M = 20. T = 10. So, 10/J + 10/J = 1/2 and J = 40. Eureka!

Answer C

I would solve this as follows:

Let Mac finish the job in M days. Thus, his rate = 1/M
Let Jack finish the job in J days. His rate = 1/J

Now, Mac leaves after working together for 'T' days and for the remaining R days Jack worked to complete the entire job.

Thus, we conclude that Mac worked for 'T' days and Jack worked for 'T+R' days.
Also, it is given that both did equal amount of work

Thus, part of job Mac did T*(1/M) = T/M ........ (1)
and Jack did (T+R)(1/J) = (T+R)/J .............. (2)

As both of them did equal amount of work, from eqn(1) above T/M = 1/2 => T= M/2 .....(3)
and from eqn(2) above, (T+R)/J = 1/2 => (T+R)= J/2 => T = J/2 - R ...................(4)

Compare value of 'T' from eqns(3) & (4), we get

M/2 = J/2 - R

We make Jack 'J' as subject of the eqn as we have to find 'the no of days Jack takes to complete the job'.

Thus, J = 2R + M .........................(5)

Now, from the question, Option(1) gives M=20 days
Eqn(5) gives us J=2R + 20
Not Sufficient as we cannot solve this eqn.

Option (2) gives R=10 days.
Not sufficient as J=20 + M does not solve the eqn.

Taking both the options together, we can substitute the value of M & R in eqn(5),
Thus we get,
J = 20 + 20 = 40 days

SUFFICIENT

[spoiler]Ans: (C)[/spoiler]