A town library allows membership to only the residents of that town.

Question

A town library allows membership to only the residents of that town. At a certain point in time all the books are
lent out. No two people have borrowed exactly the same number of books. 238 people haven’t borrowed anything
and the total number of books in the library was 105.

(i) What is the maximum number of people in the town?
(ii) What is the minimum number of people in the town?

Please help.

AndrewN · Answer

Hello,  JKruger . This is not a homework problem, is it? I ask because this is definitely NOT a GMAT™-like question. DS statements offer information that may prove sufficient to answer the question asked. You have simply presented an open-ended problem with two completely different questions. Also, I doubt the GMAT™ would use different terms— residents  and  people —to refer to the same group. If we allow for the two terms to be used interchangeably in the question, then we can proceed as follows.

The key to answering this one lies in the third line of the problem:  No two people have borrowed exactly the same number of books.  If all 105 books were lent out, then we just have to figure out the largest number of unique positive integers that can sum to 105.

1  +  2  +  3  +  4  + 5 +  6  +  7  +  8  +  9  = 45

Note that by pairing certain numbers, we keep getting sums of 10, and we are already up to 45. This idea looks promising. What if we kept going? If we examine the integer 105 itself, we can see that it factors into 3 * 5 * 7. The smallest multiple of two factors is 15, so perhaps this should be our desired sum of paired numbers rather than 10.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 = 105

There we go. At a maximum, then, we could have 14 people who borrowed the 105 books  in addition to  the 238 people who had not borrowed anything but were eligible to do so. This gives us a sum of   252 residents  .

Since we are not given any information about borrowing limits, there could be one resident who decided to check out all the books, in which case the answer would be 238 + 1, or 239 residents. Of course, since this one person had borrowed all the material, there would be nothing else for the library to lend out, and we could accurately say that, to quote the problem again,  no two people have borrowed exactly the same number of books  (i.e. 238 residents borrowed none, 1 resident borrowed all). If the question-writer means for us to interpret the third line as saying that  at least two people  borrowed books, then, since 105 is not evenly divisible by 2, there would be no way for two residents to have borrowed the same number of books. In this scenario, there would be 238 + 2, or 240 residents.

I am less satisfied with the second question than the first. I think the first could made into a decent GMAT™-like question. The second one is too open-ended. If this is all a big trap question, and  people  can refer to non-residents, then, well, what a waste of time.

- Andrew

GMATist1 · Answer

For number of people to be maximum we need borrowers to be maximum, so we need to distribute books with maximum frugality

Since none of the two borrowers can have same number of books The best possible way is: Giving 1 book to the first borrower 2 books to the second borrower 3 books to the third borrower And so on…. The pattern evident here is Arithmetic Progression with a difference of 1: 1+2+3+….+n = 105 Sum of AP = n/2 ( 2a + (n-1)d ) where a is the first term d is the difference between consecutive terms n is the total number of terms Plugging in values in the formula we get 105 = n/2( 2 + (n-1)) n = 14 Therefore maximum number of people = 14+238 = 252 For number of people to be minimum, borrowers need to be minimum. So we have only 1 borrower loaned all the books. 1+238 = 239 Posted from my mobile device

Kinshook · Answer

A town library allows membership to only the residents of that town. At a certain point in time all the books are
lent out. No two people have borrowed exactly the same number of books. 238 people haven’t borrowed anything
and the total number of books in the library was 105.

(i) What is the maximum number of people in the town?
(ii) What is the minimum number of people in the town?

Please help.

(i) What is the maximum number of people in the town?
To maximise people in the town, let us minimise the number of books borrowed per person.
Let us assume that the book borrowed = 1 + 2 + 3 + .... + n = n(n-1)/2
Sum = 105 = n(n-1)/2 
n(n-1) = 210 
n = 15
Maximum number of people in the town = 238 + 15 = 253

(ii) What is the minimum number of people in the town?
Minimum number of people in the town = 238 + 1 = 239 since only 1 person can borrow all the books of the library.

IMO D

A town library allows membership to only the residents of that town.

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