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06 Aug 2026, 10:49
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At a distribution center, stations A and B scan packages at their respective constant rates. How many more hours does it take station B, working alone, to scan a new shipment of packages than it takes station A, working alone?
(1) The ratio of the time it takes station B, working alone, to scan the shipment to the time it takes station A, working alone, to scan the shipment is \(4:1\).
(2) The ratio of the time it takes stations A and B, working together, to scan the shipment to the time it takes station A, working alone, to scan the shipment is \(4:5\).
(1) The ratio of the time it takes station B, working alone, to scan the shipment to the time it takes station A, working alone, to scan the shipment is \(4:1\).
(2) The ratio of the time it takes stations A and B, working together, to scan the shipment to the time it takes station A, working alone, to scan the shipment is \(4:5\).
06 Aug 2026, 10:49
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Official Solution:
At a distribution center, stations A and B scan packages at their respective constant rates. How many more hours does it take station B, working alone, to scan a new shipment of packages than it takes station A, working alone?
Let \(a\) be the number of hours station A takes to scan the shipment alone, and let \(b\) be the number of hours station B takes to scan the shipment alone. The question asks for \(b - a\).
(1) The ratio of the time it takes station B, working alone, to scan the shipment to the time it takes station A, working alone, to scan the shipment is \(4:1\).
So:
\(b = 4a\)
This gives only the relationship between the two times, not the actual number of hours.
Not sufficient.
(2) The ratio of the time it takes stations A and B, working together, to scan the shipment to the time it takes station A, working alone, to scan the shipment is \(4:5\).
Station A’s rate is \(\frac{1}{a}\), and station B’s rate is \(\frac{1}{b}\). Thus, their combined rate is:
\(\frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}\)
So the time they take together is:
\(\frac{ab}{a + b}\)
Statement (2) tells us that the ratio of the together time to station A’s alone time is \(4:5\):
\(\frac{\frac{ab}{a + b}}{a} = \frac{4}{5}\)
\(\frac{b}{a + b} = \frac{4}{5}\)
\(5b = 4a + 4b\)
\(b = 4a\)
This gives the same relationship as statement (1), but not the actual value of \(a\) or \(b\).
Not sufficient.
(1)+(2) Each statement gives the same relationship:
\(b = 4a\)
Thus, even after combining the statements, we still know only the relationship between the two times, not the actual value of either time. Therefore, \(b - a\) cannot be determined.
Not sufficient.
Answer: E
At a distribution center, stations A and B scan packages at their respective constant rates. How many more hours does it take station B, working alone, to scan a new shipment of packages than it takes station A, working alone?
Let \(a\) be the number of hours station A takes to scan the shipment alone, and let \(b\) be the number of hours station B takes to scan the shipment alone. The question asks for \(b - a\).
(1) The ratio of the time it takes station B, working alone, to scan the shipment to the time it takes station A, working alone, to scan the shipment is \(4:1\).
So:
\(b = 4a\)
This gives only the relationship between the two times, not the actual number of hours.
Not sufficient.
(2) The ratio of the time it takes stations A and B, working together, to scan the shipment to the time it takes station A, working alone, to scan the shipment is \(4:5\).
Station A’s rate is \(\frac{1}{a}\), and station B’s rate is \(\frac{1}{b}\). Thus, their combined rate is:
\(\frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}\)
So the time they take together is:
\(\frac{ab}{a + b}\)
Statement (2) tells us that the ratio of the together time to station A’s alone time is \(4:5\):
\(\frac{\frac{ab}{a + b}}{a} = \frac{4}{5}\)
\(\frac{b}{a + b} = \frac{4}{5}\)
\(5b = 4a + 4b\)
\(b = 4a\)
This gives the same relationship as statement (1), but not the actual value of \(a\) or \(b\).
Not sufficient.
(1)+(2) Each statement gives the same relationship:
\(b = 4a\)
Thus, even after combining the statements, we still know only the relationship between the two times, not the actual value of either time. Therefore, \(b - a\) cannot be determined.
Not sufficient.
Answer: E
