ern5231,A can do 3 dozens per hour and b can do 4 dozens per hour. At any time atleast one person is working. Both worked how many hours simultaneously to guarantee that 77 dozens are done in 14 hours?
I'm a little confused with the question, seems like there is something missing with regards to maximizing/minimizing the number of hours both A&B work together. In any case, we know that if A&B worked together for 11 hours, they would do 77 dozen, but since the question is asking to guarantee that the work be done in 14 hours, we know that somewhere in the process, from start to finish, either A or B will not work for part of it. So, here's my attempt at it:
Let the "A" be the number of hours A works alone, "B" be the number of hours B works alone, and "X" be the number of hours both A & B work together. Per the question there are no gaps while the work is being done, i.e., someone is always working. So, we can rephrase the question to read,
3A+4B+(3+4)X=77 dozen
3A+4B+7X=77.....(i)
From the question we also know that A+B+X=14 hours. Since A is the slower worker of the two, let's assume that A does not do any work by himself, but instead joins B at some point in time (still agrees with the question since "at any time atleast one person is working"). I am assuming so, because I am trying to solve for how long A&B have to work for at the least to guarantee that the work gets done in 14 hrs. So, I'm ignoring the slowest worker. Therefore, A works 0 hours by himself. So, we can rephrase the above equation as:
B+X=14 --> B=14-X.....(ii)
Combining the two equations,
3A+4(14-X)+7x=77
56-4X+7X=77 (as I assumed above, A will work 0 hours by himself)
3X=21
X=7
Therefore, A&B need to work for 7 hours at least to guarantee that the work gets done in 14 hours.
Hope that helped - like I said, I'm just taking a shot at it. By the way, what is the source for this question, also what's the official answer?
-Ash
