Bunuel wrote:

A and B working separately can do a piece of work in 9 and 12 days respectively.If they work for a day alternately, A beginning, in how many days, the work will be completed?

A. \(10 \frac{1}{5}\)

B. \(10 \frac{1}{4}\)

C. \(10 \frac{1}{3}\)

D. \(10 \frac{1}{2}\)

E. \(10 \frac{2}{3}\)

We see that A’s rate = 1/9, and B’s rate = 1/12. Since A starts first, A works one more day than B. So if we let x = the number of days B works, then x + 1 = the number of days A works. We can create the equation:

(1/9)(x + 1) + (1/12)x = 1

Multiplying both sides by 36, we have:

4(x + 1) + 3x = 36

4x + 4 + 3x = 36

7x = 32

x = 32/7 = 4 4/7

We see that x is not a whole number. If we round x up to 5, then certainly the two workers will be “overworked.” That is, the amount they work together will exceed 1 job. Therefore, we have to round x down to 4. That is, B works 4 days and A works 5 days. They will now be “underworked,” but we can figure out how much time will be needed for the remainder of the job. So if B works 4 days and A works 5 days, together they have completed 4(1/12) + 5(1/9) = 1/3 + 5/9 = 8/9 of the job. If the remaining 1/9 of the job will be completed by B for one more day, then 1/9 - 1/12 = 1/36 of the job still needs to be completed and that fraction of the job can be completed by A in (1/36)/(1/9) = 9/36 = 1/4 of a day. Therefore, in total, they work 4 + 5 + 1 + 1/4 = 10 1/4 days.

Alternate Solution:

We can solve this problem by arithmetic. Let’s pair two days together: an “AB” pair is comprised of one day worked by A and the next day worked by B. The combined work in one AB pair is:

1/9 + 1/12 = 7/36 of the job.

Since the entire job is 1, or 36/36, we see that if we have 5 “AB” paired days (for a total of 10 days, 5 days worked by A and 5 worked by B), we will have finished 5 x 7/36, or 35/36, of the total job. Thus, we still have 1/36 of the job to complete. Since it is now A’s turn to work, we see that A can finish the remaining 1/36 of the job in (1/36)/(1/9) = 1/4 of a day. Thus, the total time for the job to be completed is 10 1/4 days.

Answer: B

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