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30 May 2026, 08:32
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A factory recently bought two machines, A and B, each operating at its own constant rate. Working alone, Machine A takes \(n\) minutes to complete a Junior order. Working alone, Machine B takes \(m\) minutes to complete a Jumbo order, which is exactly 5 times the size of a Junior order. Is \(n\) greater than \(m\)?
(1) Working alone on a Junior order, Machine B takes half as many minutes as Machine A.
(2) Machine B’s work rate is more than \(\frac{3}{2}\) times Machine A’s work rate.
(1) Working alone on a Junior order, Machine B takes half as many minutes as Machine A.
(2) Machine B’s work rate is more than \(\frac{3}{2}\) times Machine A’s work rate.
30 May 2026, 08:32
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Official Solution:
A factory recently bought two machines, A and B, each operating at its own constant rate. Working alone, Machine A takes \(n\) minutes to complete a Junior order. Working alone, Machine B takes \(m\) minutes to complete a Jumbo order, which is exactly 5 times the size of a Junior order. Is \(n\) greater than \(m\)?
Treat 1 Junior order as 1 unit of work. Then 1 Jumbo order is 5 units of work.
• Machine A completes 1 unit in \(n\) minutes, so its rate is \(\frac{1}{n}\) units per minute.
• Machine B completes 5 units in \(m\) minutes, so its rate is \(\frac{5}{m}\) units per minute.
We need to determine whether \(n > m\), or which is the same whether \(\frac{n}{m} > 1\).
(1) Working alone on a Junior order, Machine B takes half as many minutes as Machine A.
If Machine B takes half as many minutes as Machine A to complete a Junior order, then Machine B completes 1 unit in \(\frac{n}{2}\) minutes, so its rate is \(\frac{2}{n}\) units per minute. Since Machine B’s rate is also \(\frac{5}{m}\), we have \(\frac{5}{m} = \frac{2}{n}\), so \(\frac{n}{m} = \frac{2}{5}\), giving a No answer to the question. Sufficient.
(2) Machine B’s work rate is more than \(\frac{3}{2}\) times Machine A’s work rate.
This implies \(\frac{5}{m} > \frac{3}{2} * \frac{1}{n}\), so \(\frac{n}{m} > \frac{3}{10}\). Thus, \(\frac{n}{m}\) may or may not be greater than 1. Not sufficient.
Answer: A
A factory recently bought two machines, A and B, each operating at its own constant rate. Working alone, Machine A takes \(n\) minutes to complete a Junior order. Working alone, Machine B takes \(m\) minutes to complete a Jumbo order, which is exactly 5 times the size of a Junior order. Is \(n\) greater than \(m\)?
Treat 1 Junior order as 1 unit of work. Then 1 Jumbo order is 5 units of work.
• Machine A completes 1 unit in \(n\) minutes, so its rate is \(\frac{1}{n}\) units per minute.
• Machine B completes 5 units in \(m\) minutes, so its rate is \(\frac{5}{m}\) units per minute.
We need to determine whether \(n > m\), or which is the same whether \(\frac{n}{m} > 1\).
(1) Working alone on a Junior order, Machine B takes half as many minutes as Machine A.
If Machine B takes half as many minutes as Machine A to complete a Junior order, then Machine B completes 1 unit in \(\frac{n}{2}\) minutes, so its rate is \(\frac{2}{n}\) units per minute. Since Machine B’s rate is also \(\frac{5}{m}\), we have \(\frac{5}{m} = \frac{2}{n}\), so \(\frac{n}{m} = \frac{2}{5}\), giving a No answer to the question. Sufficient.
(2) Machine B’s work rate is more than \(\frac{3}{2}\) times Machine A’s work rate.
This implies \(\frac{5}{m} > \frac{3}{2} * \frac{1}{n}\), so \(\frac{n}{m} > \frac{3}{10}\). Thus, \(\frac{n}{m}\) may or may not be greater than 1. Not sufficient.
Answer: A
