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?Say k, m, and p are the numbers of minutes machines K, M, and P take, respectively, to complete the task. Then we have that \(\frac{1}{k}+\frac{1}{m}+\frac{1}{p}=\frac{1}{24}\).
(1) Machines M and P, working simultaneously and independently at their respective constant rates, can complete the task in 36 minutes --> \(\frac{1}{m}+\frac{1}{p}=\frac{1}{36}\), thus \(\frac{1}{k}+\frac{1}{36}=\frac{1}{24}\) --> we can find the value of \(k\). Sufficient.
(2) Machines K and P, working simultaneously and independently at their respective constant rates, can complete the task in 48 minutes --> \(\frac{1}{k}+\frac{1}{p}=\frac{1}{48}\). The value of k cannot be determined from the data we have. Not sufficient.
Answer: A.
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