Walkabout wrote:
Three machines, K, M, and P, working simultaneously and independently at their respective constant rates, can complete a certain task in 24 minutes. How long does it take Machine K, working alone at its constant rate, to complete the task?
(1) Machines M and P, working simultaneously and independently at their respective constant rates, can complete the task in 36 minutes.
(2) Machines K and P, working simultaneously and independently at their respective constant rates, can complete the task in 48 minutes.
We are given that machines K, M, and P, working simultaneously and independently at their respective constant rates, can complete a certain task in 24 minutes. If we consider the entire task to be equal to 1, and the time in minutes for machines K, M, and P to complete the task to be k, m, and p, respectively, then the rates of machines K, M, and P are:
1/k = rate of machine K to complete the task
1/m = rate of machine M to complete the task
1/p = rate of machine P to complete the task
Since it takes machines K, M, and P, working simultaneously and independently, 24 minutes, the combined rate of machines K, M, and P is 1 task per 24 minutes. That is:
1/k + 1/m + 1/p = 1/24
We need to determine how long it takes machine K to complete the task, or in other words, the value of k. Since 1/k + 1/m + 1/p = 1/24, the rate of machine K is:
1/k = 1/24 - 1/m - 1/p
1/k = 1/24 - (1/m + 1/p)
Thus, if we can determine the value of (1/m + 1/p), we can determine the value of 1/k and hence the value of k.
Statement One Alone:
Machines M and P, working simultaneously and independently at their respective constant rates, can complete the task in 36 minutes.
From statement one we know:
1/m + 1/p = 1/36
Thus, the rate for machine K to complete the task is 1/24 - 1/36 = 3/72 - 2/72 = 1/72, and therefore, the time for machine K to complete the task is 72 minutes.
Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.
Statement Two Alone:
Machines K and P, working simultaneously and independently at their respective constant rates, can complete the task in 48 minutes.
From statement two we know:
1/k + 1/p = 1/48
Since we don’t know the value of p, this is not enough information to determine the value of k.
Statement two alone is not sufficient to answer the question.
Answer: A