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At a certain library, 12 students borrowed a total of

by M7MBA » Thu May 31, 2018 1:48 am

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At a certain library, 12 students borrowed a total of 71 books. If the range of the number of books borrowed per student was 5, which of the following CANNOT be the number of students who borrowed 6 books?

I. 9
II. 10
III. 11

A. I only
B. III only
C. II and III only
D. I and III only
E. I, II and III

The OA is the option B.

Is there a strategic approach here? I'd really appreciate any help that anyone can give me. <i class="em em-confused"></i>
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by GMATGuruNY » Thu May 31, 2018 3:59 am
M7MBA wrote:At a certain library, 12 students borrowed a total of 71 books. If the range of the number of books borrowed per student was 5, which of the following CANNOT be the number of students who borrowed 6 books?

I. 9
II. 10
III. 11

A. I only
B. III only
C. II and III only
D. I and III only
E. I, II and III
Statement I: 9 students borrow 6 books each
Let the other 3 students be A, B and C.
Since 9 students borrow 6 books each, the list of borrowed books is as follows:
A, B, C, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6.
Books borrowed by A+B+C = (total books borrowed) - (books borrowed by the 9 students in blue) = 71 - (9*6) = 17.
Since A+B+C = 17, here is one way a range of 5 can be yielded:
A=4, B=4, 6, 6, 6, 6, 6, 6, 6, 6, 6, C=9.
Since Statement I is possible, the correct answer cannot include I.
Eliminate A, D and E.

Statement II: 10 students borrow 6 books each
Let the other 2 students be A and B.
Since 10 students borrow 6 books each, the list of borrowed books is as follows:
A, B, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6.
Books borrowed by A+B = (total books borrowed) - (books borrowed by the 10 students in blue) = 71 - (10*6) = 11.
Since A+B=11, a range of 5 can be yielded as follows:
A=3, 6, 6, 6, 6, 6, 6, 6, 6, 6, B=8.
Since Statement II is possible, the correct answer cannot include II.
Eliminate C.

The correct answer is B.
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by swerve » Thu May 31, 2018 10:23 am
x1 + x2 + .... + x12 (in ascending order) = 71

Range = 5

Question: which of the following CANNOT be the number of students who borrowed 6 books

Start from high to low

III) Number of students who borrowed 6 books = 11 (Assume x2.....x12)--> x1 + 11 * 6 = 71 --> x1 = 5. Here Range is 1. Not possible.

II) Number of students who borrowed 6 books = 10 (Assume x2.....x11) --> x1 + 10 * 6 + x12 = 71 --> x1 + x12 = 11 --> x1 = 3 and x12 = 8.
Range = 5. Possible

I) Number of students who borrowed 6 books = 9 (Assume x3......x11) --> x1 + x2 + 9*6 + x3 = 71 --> x1 = x2 = 4 and x3 = 9.
Range = 5. Possible

Option B is the correct answer.

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by Jeff@TargetTestPrep » Fri Jun 01, 2018 11:58 am
M7MBA wrote:At a certain library, 12 students borrowed a total of 71 books. If the range of the number of books borrowed per student was 5, which of the following CANNOT be the number of students who borrowed 6 books?

I. 9
II. 10
III. 11

A. I only
B. III only
C. II and III only
D. I and III only
E. I, II and III
Let's analyze each Roman Numeral.

I. 9

If 9 people borrowed 6 books, we have 71 - 54 = 17 books for the remaining 3 people. If we let x = the least number of books borrowed by 1 of the 3 people, then x + 5 = the greatest number of books borrowed by another person.

Let y = the number of books borrowed by the last person.

Notice that 0 ≤ x ≤ y ≤ x + 5 and none of x, y, x + 5 can be 6. So we have:

x + x + 5 + y = 17

2x + y = 12

We see that x can't be 7 or more (otherwise, y will be negative).

If x = 5, then y = 2 and x + 5 = 10. But the range is 10 - 2 = 8, not 5.

If x = 4, then y = 4 and x + 5 = 9. We see that 4 + 4 + 9 = 17 and the range is 9 - 4 = 5.
So it's possible to have 9 people who borrowed 6 books each.

II. 10

If 10 people borrowed 6 books, we have 71 - 60 = 11 books for the remaining 2 people.

If we let x = the least number of books borrowed by 1 of the 2 people, then x + 5 = the most number of books borrowed by the other person.

Notice neither x nor x + 5 can be 6. So we have:

x + x + 5 = 11

2x = 6

x = 3

We see that the range is 3 + 8 = 11 and the range is 8 - 3 = 5. So it's possible to have 10 people who borrowed 6 books each.

III. 11

If 11 people borrowed 6 books, we have 71 - 66 = 5 books for the remaining person. But then the range is 6 - 5 = 1, not 5. So 11 CAN'T be the number of students who borrowed 6 books each.

Answer: B

Jeffrey Miller
Head of GMAT Instruction
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