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30 May 2026, 10:14
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An inspection agency follows these guidelines for newly laid pipelines: all of the pipeline up to kilometer 60 is to be inspected; in addition, \(\frac{2}{3}\) of any portion of the pipeline between kilometer 60 and kilometer 210 that exists is to be inspected; and no part of the pipeline beyond kilometer 210 is to be inspected. The agency was assigned to inspect a newly laid pipeline in accordance with these guidelines. How long was the pipeline?
(1) A total of 160 kilometers of the pipeline was inspected.
(2) The length of the pipeline that was inspected was at least \(\frac{16}{21}\) of the total length of the pipeline.
(1) A total of 160 kilometers of the pipeline was inspected.
(2) The length of the pipeline that was inspected was at least \(\frac{16}{21}\) of the total length of the pipeline.
30 May 2026, 10:14
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Official Solution:
A pipeline inspection team was assigned to inspect a newly laid pipeline. According to the inspection plan, all of the first 60 kilometers of the pipeline were inspected, \(\frac{2}{3}\) of the next 150 kilometers were inspected, and no part of the pipeline beyond the first 210 kilometers was inspected. How long was the pipeline?
(1) A total of 160 kilometers of the pipeline was inspected.
The maximum possible inspected length is
\(60 + \frac{2}{3} * 150 = 160\)
So for 160 kilometers to have been inspected, the pipeline must have been at least 210 kilometers long.
Not sufficient.
(2) The length of the pipeline that was inspected was at least \(\frac{16}{21}\) of the total length of the pipeline.
From the inspection plan, the greatest possible inspected length is
\(60 + \frac{2}{3} * 150 = 160\)
If the pipeline had been longer than 210 kilometers, the fraction inspected would have been \(\frac{160}{\text{more than }210}\), which is less than \(\frac{160}{210} = \frac{16}{21}\). Since (2) says that the fraction inspected was at least \(\frac{16}{21}\), the pipeline must have been no more than 210 kilometers long. Not sufficient.
(1) + (2):
From (1), the pipeline was at least 210 kilometers long.
From (2), the pipeline was no more than 210 kilometers long.
Therefore, the pipeline was exactly 210 kilometers long.
Sufficient.
Answer: C
A pipeline inspection team was assigned to inspect a newly laid pipeline. According to the inspection plan, all of the first 60 kilometers of the pipeline were inspected, \(\frac{2}{3}\) of the next 150 kilometers were inspected, and no part of the pipeline beyond the first 210 kilometers was inspected. How long was the pipeline?
(1) A total of 160 kilometers of the pipeline was inspected.
The maximum possible inspected length is
\(60 + \frac{2}{3} * 150 = 160\)
So for 160 kilometers to have been inspected, the pipeline must have been at least 210 kilometers long.
Not sufficient.
(2) The length of the pipeline that was inspected was at least \(\frac{16}{21}\) of the total length of the pipeline.
From the inspection plan, the greatest possible inspected length is
\(60 + \frac{2}{3} * 150 = 160\)
If the pipeline had been longer than 210 kilometers, the fraction inspected would have been \(\frac{160}{\text{more than }210}\), which is less than \(\frac{160}{210} = \frac{16}{21}\). Since (2) says that the fraction inspected was at least \(\frac{16}{21}\), the pipeline must have been no more than 210 kilometers long. Not sufficient.
(1) + (2):
From (1), the pipeline was at least 210 kilometers long.
From (2), the pipeline was no more than 210 kilometers long.
Therefore, the pipeline was exactly 210 kilometers long.
Sufficient.
Answer: C
