Problem Solving

1. Time taken by A to complete a job is 6 days. Time taken by B to complete the same job is 12 days. They start working together and work for 2 days. B then leaves and A is left alone to complete the job. In how many days will A finish the remaining work, if it is given that A's efficiency decreases by 25 percent after B left?
A. 8
B. 10
C. 12
D. 14
E. 16
oaE

raj44 wrote:1. Time taken by A to complete a job is 6 days. Time taken by B to complete the same job is 12 days. They start working together and work for 2 days. B then leaves and A is left alone to complete the job. In how many days will A finish the remaining work, if it is given that A's efficiency decreases by 25 percent after B left?
A. 8
B. 10
C. 12
D. 14
E. 16

Let the job = 24 widgets.
Since A takes 6 days to produce 24 widgets, A's rate = w/t = 24/6 = 4 widgets per day.
Since B takes 12 days to produce 24 widgets, B's rate = w/t = 24/12 = 2 widgets per day.

Combined rate for A and B working together = 4+2 = 6 widgets per day.
In 2 days, the amount of work produced by A and B = r*t = 2*6 = 12 widgets.
Remaining work = 24-12 = 12 widgets.

A's rate reduced by 25% = 4 - (0.25)4 = 3 widgets per day.
At a rate of 3 widgets per day, the time for A to produce the remaining 12 widgets = w/r = 12/3 = 4 days.

The OA is incorrect.
The required time is not among the answer choices.
What is the source of this problem?

Correct me if im wrong, but just looking at the problem. With two people working on a job together, the job should take less than than anyone of them individually. In this case, A is most efficient, so the answer should be less than A's 6 days per job.

sn0rky wrote:With two people working on a job together, the job should take less than than anyone of them individually.

Yes -- if none of the rates change over the course of the problem.
But in the problem above, A's rate decreases.

In my solution above, A and B work together to produce 12 of the 24 widgets.
If, following B's departure, A's rate were to decrease by 75% to 1 widget per day, then A would require 12 days to complete the remaining 12 widgets -- twice the number of days for him to complete the entire job at his regular rate.

Remember rate questions use reciprocals:
A's rate = 1/6
B's rate = 1/12
Joint rate = 1/6 + 1/12 = 1/4
2 days at 1/4 = 2 x 1/4 = 1/2 -> job is half complete
A's new rate = 0.75 x 1/6 = 1/8 to do the other half of the job
so A will take 8/2 = 4 days

In 2 days: at current rates A does 1/3 of the job and B does 1/6 of the job ~ so 2/6 + 1/6 = 3/6 = 1/2 of the job is complete.
A's new rate after 2 days: 6 days / 0.75 = 8 days.
A needs 4 days to complete the job.

mbawisdom wrote:In 2 days: at current rates A does 1/3 of the job and B does 1/6 of the job ~ so 2/6 + 1/6 = 3/6 = 1/2 of the job is complete.
A's new rate after 2 days: 6 days * 0.75 = 4.5 days.
A needs 2.25 days to complete the job.

Answer not among the ones given.

The portion in red is incorrect.

A's time to produce the entire job = 6 days.
Rate and time are RECIPROCALS.
If A works at 3/4 his normal rate, he will take 4/3 of his normal time:
(4/3)(6) = 8 days for A to produce the entire job.

Thus, to produce the remaining half of the job after B's departure, A will require 4 days.

Yuck, lots of steps!

A's rate = 1/6
B's rate = 1/12
Joint rate = 1/6 + 1/12 = 1/4

Working together for two days, A and B do 2 * (1/4) or 1/2 of the job. That means

Remaining Work = 1/2
A's reduced rate = (3/4)(1/6) = 1/8

W = RT, so 1/2 = 1/8 * T, so T = 4. As noted, the OA is incorrect: the time to complete the job is 4 days.

HOWEVER, if we assume that A's efficiency decreases by 25% PER DAY, we'd have

(3/4)(1/6) + (3/4)²(1/6) + (3/4)³(1/6) + ... + (3/4)�(1/6) = 1/2, or
(3/4) + (3/4)² + (3/4)³ + ... (3/4)� = 3

The GMAT wouldn't assume that we know how to solve this, but in 16 days we'd be almost done. (Unfortunately it would take us an INFINITE number of days to get to 3 ... so "almost" is relative. )