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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # At a certain library, 12 students borrowed a total of 71 books. If the

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Math Expert V
Joined: 02 Sep 2009
Posts: 59590
At a certain library, 12 students borrowed a total of 71 books. If the  [#permalink]

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13 00:00

Difficulty:   95% (hard)

Question Stats: 39% (02:19) correct 61% (02:59) wrong based on 127 sessions

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GMAT CLUB'S FRESH CHALLENGE QUESTION:

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

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Marshall & McDonough Moderator D
Joined: 13 Apr 2015
Posts: 1682
Location: India
Re: At a certain library, 12 students borrowed a total of 71 books. If the  [#permalink]

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3
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

Target Test Prep Representative G
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2809
Re: At a certain library, 12 students borrowed a total of 71 books. If the  [#permalink]

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Bunuel wrote:

GMAT CLUB'S FRESH CHALLENGE QUESTION:

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.

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Math Expert V
Joined: 02 Sep 2009
Posts: 59590
Re: At a certain library, 12 students borrowed a total of 71 books. If the  [#permalink]

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Bunuel wrote:

GMAT CLUB'S FRESH CHALLENGE QUESTION:

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

_________________
SVP  D
Joined: 03 Jun 2019
Posts: 1879
Location: India
Re: At a certain library, 12 students borrowed a total of 71 books. If the  [#permalink]

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Bunuel wrote:

GMAT CLUB'S FRESH CHALLENGE QUESTION:

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

Given; At a certain library, 12 students borrowed a total of 71 books.

Asked; 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?

Let x1 + x2 + ....+ x12 = 71

I. 9
x1 + x2 + ... + x9 = 6*9 = 54
x10 + x11 + x12 = 71 - 54 = 17
If x10 = x11 = 4 and x12 = 9
Possible
II. 10
x1 + x2 + .... + x10 = 6*10 = 60
x11 + x12 = 11
x11 = 3; x12 = 8
Possible
III. 11
x1 + ... + x11 = 11*6 = 66
x12 = 71 -66 = 5
Not possible

IMO B Re: At a certain library, 12 students borrowed a total of 71 books. If the   [#permalink] 08 Oct 2019, 01:46
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# At a certain library, 12 students borrowed a total of 71 books. If the  