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For a certain city's library, the average cost of purchasing each new book is \(\$28\). The library receives \(\$15{,}000\) from the city each year, the library also receives a bonus of \(\$2{,}000\) if the total number of items checked out over the course of the year exceeds \(5{,}000\). Did the library receive the bonus last year?
Let's break down what we're being asked: Did the library receive the bonus last year?
This is a yes/no question. We need to determine whether we can definitively answer YES or NO.
To answer this question, we need to know whether the total number of items checked out last year was greater than 5,000. If yes → library received bonus. If no → library didn't receive bonus.
The question becomes sufficient when we can determine with certainty whether checkouts > 5,000.
Statement 1: The library purchased an average of 50 new books each month last year and received enough money from the city to cover this cost.
Let's calculate the total book purchasing cost:
Now here's the key insight: The statement says the city provided "enough money to cover this cost."
If the library only received the base \(\$15,000\):
But we're told the city DID provide enough money. The only way this is possible is if the library received the bonus:
Critical conclusion: Since the library received the bonus, they MUST have checked out more than 5,000 items (that's the only way to earn the bonus).
[STOP - Sufficient!] Statement 1 allows us to answer YES, the library received the bonus.
This eliminates choices B, C, and E.
Important: Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: The lowest number of items checked out in one month was 459.
If the minimum monthly checkouts was 459, then we can determine the minimum yearly total:
Since \(5,508 > 5,000\), the library definitely exceeded the threshold required for the bonus.
Think of it this way: Even in their worst month, they checked out 459 items. So their yearly total must be at least \(459 \times 12 = 5,508\), which exceeds the 5,000 threshold.
[STOP - Sufficient!] Statement 2 allows us to answer YES, the library received the bonus.
This eliminates choices A, C, and E.
Both statements independently allow us to determine that the library received the bonus, though through completely different logical paths:
Answer Choice D: "Each statement alone is sufficient."