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At the Northgate distribution center, a sorting line can run in one of two modes: automated mode, in which it processes 180 parcels per hour, and manual-check mode, in which it processes 45 parcels per hour. For a certain batch, was the line’s average (arithmetic mean) processing rate greater than 120 parcels per hour?
(1) The batch contained 240 parcels.
(2) More than \(\frac{3}{4}\) of the parcels in the batch were processed in automated mode.
(1) The batch contained 240 parcels.
(2) More than \(\frac{3}{4}\) of the parcels in the batch were processed in automated mode.
30 May 2026, 10:40
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Official Solution:
At the Northgate distribution center, a sorting line can run in one of two modes: automated mode, in which it processes 180 parcels per hour, and manual-check mode, in which it processes 45 parcels per hour. For a certain batch, was the line’s average (arithmetic mean) processing rate greater than 120 parcels per hour?
Let the total number of parcels be \(n\), and suppose a fraction \(x\) of the parcels was processed in automated mode. Then \(1 - x\) was processed in manual-check mode.
The total time would then be \(\frac{\text{work}}{\text{rate}}\):
\(\frac{nx}{180} + \frac{n(1 - x)}{45}\)
So the average processing rate would be \(\frac{\text{work}}{\text{total time}}\):
\(\frac{n}{\frac{nx}{180} + \frac{n(1 - x)}{45}}\)
\(= \frac{1}{\frac{x}{180} + \frac{1 - x}{45}}\)
\(= \frac{1}{\frac{x + 4 - 4x}{180}}\)
\(= \frac{1}{\frac{4 - 3x}{180}}\)
\(= \frac{180}{4 - 3x}\)
We are asked whether this is greater than 120:
Is \(\frac{180}{4 - 3x} > 120\)?
Since \(0 \leq x \leq 1\), we have \(4 - 3x > 0\), so we can multiply both sides without changing the direction of the inequality:
Is \(180 > 120(4 - 3x)\)?
Is \(180 > 480 - 360x\)?
Is \(360x > 300\)?
Is \(x > \frac{5}{6}\)?
Thus, the question essentially asks whether the fraction of parcels processed in automated mode was greater than \(\frac{5}{6}\) of the total.
(1) The batch contained 240 parcels.
This gives only the total number of parcels, which does not determine whether \(x > \frac{5}{6}\). Not sufficient.
(2) More than \(\frac{3}{4}\) of the parcels in the batch were processed in automated mode.
Since \(\frac{3}{4}\) is less than \(\frac{5}{6}\), the fraction processed in automated mode could still have been less than \(\frac{5}{6}\) or greater than \(\frac{5}{6}\). Not sufficient.
(1)+(2) Statement (1) still adds nothing useful, and statement (2) still does not tell us whether \(x > \frac{5}{6}\). Not sufficient.
Answer: E
At the Northgate distribution center, a sorting line can run in one of two modes: automated mode, in which it processes 180 parcels per hour, and manual-check mode, in which it processes 45 parcels per hour. For a certain batch, was the line’s average (arithmetic mean) processing rate greater than 120 parcels per hour?
Let the total number of parcels be \(n\), and suppose a fraction \(x\) of the parcels was processed in automated mode. Then \(1 - x\) was processed in manual-check mode.
The total time would then be \(\frac{\text{work}}{\text{rate}}\):
\(\frac{nx}{180} + \frac{n(1 - x)}{45}\)
So the average processing rate would be \(\frac{\text{work}}{\text{total time}}\):
\(\frac{n}{\frac{nx}{180} + \frac{n(1 - x)}{45}}\)
\(= \frac{1}{\frac{x}{180} + \frac{1 - x}{45}}\)
\(= \frac{1}{\frac{x + 4 - 4x}{180}}\)
\(= \frac{1}{\frac{4 - 3x}{180}}\)
\(= \frac{180}{4 - 3x}\)
We are asked whether this is greater than 120:
Is \(\frac{180}{4 - 3x} > 120\)?
Since \(0 \leq x \leq 1\), we have \(4 - 3x > 0\), so we can multiply both sides without changing the direction of the inequality:
Is \(180 > 120(4 - 3x)\)?
Is \(180 > 480 - 360x\)?
Is \(360x > 300\)?
Is \(x > \frac{5}{6}\)?
Thus, the question essentially asks whether the fraction of parcels processed in automated mode was greater than \(\frac{5}{6}\) of the total.
(1) The batch contained 240 parcels.
This gives only the total number of parcels, which does not determine whether \(x > \frac{5}{6}\). Not sufficient.
(2) More than \(\frac{3}{4}\) of the parcels in the batch were processed in automated mode.
Since \(\frac{3}{4}\) is less than \(\frac{5}{6}\), the fraction processed in automated mode could still have been less than \(\frac{5}{6}\) or greater than \(\frac{5}{6}\). Not sufficient.
(1)+(2) Statement (1) still adds nothing useful, and statement (2) still does not tell us whether \(x > \frac{5}{6}\). Not sufficient.
Answer: E
