16 people can write 52 books in 12 days, working 8 hours a day. In how many…

The Men-Days-Work relationship states that, for a fixed number of working hours per day, the following holds: M1 x D1 x W2 = M2 x D2 x W1, where M is the number of people, D is the number of days, and W is the quantity of work completed (here, the number of books written).

1. From the first situation: M1 = 16 people, D1 = 12 days, W1 = 52 books.

2. From the second situation: M2 = 64 people, W2 = 206 books, and D2 is the number of days to find.

3. Since the daily working hours (8 hours) stay the same in both situations, apply M1 x D1 x W2 = M2 x D2 x W1: 16 x 12 x 206 = 64 x D2 x 52.

4. Compute the left side: 16 x 12 = 192, and 192 x 206 = 39552.

5. Compute the right-side coefficient: 64 x 52 = 3328.

6. So D2 = 39552 / 3328.

7. Divide numerator and denominator by 16: D2 = 2472 / 208.

8. Divide numerator and denominator by 8 again: D2 = 309/26 days (approximately 11.88 days).

Cross-check using the unitary work-rate method: one person's daily output is 52 / (16 x 12) = 52/192 books per day. With 64 people working D2 days, total output is 64 x D2 x 52/192, which must equal 206. Solving gives D2 = 206 x 192 / (64 x 52) = 39552/3328 = 309/26 days -- the same result, confirming the answer.