A small library has books on twenty different subjects including, most prominently, history. One afternoon, a librarian arbitrarily picks up...
GMAT Data Sufficiency : (DS) Questions
A small library has books on twenty different subjects including, most prominently, history. One afternoon, a librarian arbitrarily picks up 100 books for reshelving from among the books returned that day. All the books returned that day were borrowed during the past three weeks. Are fewer than 20 percent of the books the librarian picks up on history?
- Fewer than 20 percent of the books in the library are on history.
- During the past three weeks, exactly 15 books on history were borrowed from the library.
Understanding the Question
We need to determine if fewer than 20% of the 100 books picked by the librarian are history books. Since the librarian picks exactly 100 books, this translates to: Are there fewer than 20 history books among the 100 picked?
Given Information
- The library has books on 20 different subjects
- History is the most prominent subject
- The librarian arbitrarily picks 100 books from those returned that day
- All returned books were borrowed during the past 3 weeks
What We Need to Determine
This is a yes/no question. We need a definitive answer:
- YES if there are fewer than 20 history books in the selection
- NO if there are 20 or more history books in the selection
For sufficiency, we need information that allows us to answer this question with certainty.
Analyzing Statement 1
Statement 1: "Fewer than 20% of the books in the library are on history."
This might seem helpful, but here's the crucial insight: the librarian isn't picking from all library books—she's picking from books that were returned that day, which were all borrowed in the past 3 weeks.
Why Library Composition Doesn't Determine Borrowing Patterns
The overall library composition tells us nothing definitive about recent borrowing patterns. Let's see why:
Scenario A - History is trending:
- Library composition: Less than 20% history books
- Recent borrowing pattern: 40% of borrowed books were history
- Result: The 100 returned books could include 40+ history books → NO
Scenario B - History is unpopular lately:
- Library composition: Less than 20% history books
- Recent borrowing pattern: Only 5% of borrowed books were history
- Result: The 100 returned books might include only 5 history books → YES
Since we can get different answers depending on borrowing patterns, Statement 1 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
Analyzing Statement 2
Forget Statement 1. We're analyzing Statement 2 independently.
Statement 2: "During the past three weeks, exactly 15 books on history were borrowed from the library."
This statement provides a powerful constraint:
- All returned books were borrowed in the past 3 weeks
- Only 15 history books were borrowed during this entire period
- Therefore, at most 15 history books could be among the returned books
The Key Insight
Since the librarian picks from the returned books, and there are at most 15 history books available to pick, it's impossible for her to pick 20 or more history books. Even if she picked every single history book that was returned, she'd get at most 15.
Since \(15 < 20\), we can definitively answer: YES, fewer than 20% of the picked books are history books.
[STOP - Sufficient!] Statement 2 alone is sufficient.
The Answer: B
Statement 2 alone provides a definitive YES answer by establishing an absolute upper limit: at most 15 history books could be picked, which is less than 20.
Answer Choice B: Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.