For a certain city's library, the average cost of purchasing each new book is $28. The library receives $15{,}000 from...
GMAT Data Sufficiency : (DS) Questions
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?
- The library purchased an average of \(50\) new books each month last year and received enough money from the city to cover this cost.
- The lowest number of items checked out in one month was \(459\).
Understanding the Question
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.
Given Information
- Average cost per new book: \(\$28\)
- Annual funding from city: \(\$15,000\) (base amount)
- Bonus amount: \(\$2,000\)
- Bonus condition: Total items checked out must exceed 5,000
What We Need to Determine
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.
Analyzing Statement 1
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.
What This Reveals
Let's calculate the total book purchasing cost:
- Books purchased: \(50 \text{ books/month} \times 12 \text{ months} = 600 \text{ books/year}\)
- Total cost: \(600 \text{ books} \times \$28/\text{book} = \$16,800\)
Now here's the key insight: The statement says the city provided "enough money to cover this cost."
The Logical Deduction
If the library only received the base \(\$15,000\):
- \(\$15,000 < \$16,800\) (NOT enough to cover the cost) ❌
But we're told the city DID provide enough money. The only way this is possible is if the library received the bonus:
- \(\$15,000 + \$2,000 = \$17,000 > \$16,800\) ✓
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.
Analyzing Statement 2
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.
What This Means
If the minimum monthly checkouts was 459, then we can determine the minimum yearly total:
- Minimum yearly checkouts = \(459 \times 12 = 5,508 \text{ items}\)
The Clear Conclusion
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.
The Answer: D
Both statements independently allow us to determine that the library received the bonus, though through completely different logical paths:
- Statement 1: Uses the funding amount to deduce the bonus was received
- Statement 2: Uses the minimum checkout volume to prove the threshold was exceeded
Answer Choice D: "Each statement alone is sufficient."