Question

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.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is A Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

• Let k, m, and p be the numbers of minutes machines K, M, and P take, respectively, to complete the task. Then, Machine K can do of the task in 1 minute, Machine M can do of the task in 1 minute, and Machine P can do of the task in 1 minute. If all three machines working together can do the task in 24 minutes, then they can do of the task in 1 minute. So, and . Determine k.

1. Since M and P together can do the task in 36 minutes, they can do in 1 minute, and so . Then , which can be solved for a unique value of k; SUFFICIENT.
2. Since K and P together can do the task in 48 minutes, they can do in 1 minute, and so . From this information, there is no way to uniquely determine and therefore no way to uniquely determine k. For example, if p = 96, then and k = 96. On the other hand, if p = 60, then and k = 240. So, 96 and 240 are both possible values for k; NOT sufficient.

• The correct answer is A; statement 1 alone is sufficient.