Jeff Miller

GMAT OFFICIAL GUIDE DS – A certain bookcase has 2 shelves of books…

A certain bookcase has 2 shelves of books…

Solution:

We are given that a bookcase has 2 shelves of books. We are also given that the book with the greatest number of pages on the upper shelf has 400 pages and that the book with the least number of pages on the lower shelf has 475 pages. This tells us that all of the books on the upper shelf have fewer pages than all of the books on the lower shelf.

We must determine the median number of pages of the books on the two shelves. Thus, if we were to order books on both shelves from the books with the least numbers pages to the books with the greatest number of pages, we must determine the number of pages in the middle book.

Statement One Alone:

There are 25 books on the upper shelf.

Using the information in statement one, we know that the 25th book on the top shelf, when the books are ordered by the least number of pages to the greatest number of pages, has 400 pages. However, without knowing the number of books on the lower shelf, we still cannot determine the median number of pages for all the books on the two shelves.  Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

There are 24 books on the lower shelf.

Using the information in statement two, we know that the first book on the bottom shelf, when the books are ordered by the least number of pages to the greatest number of pages, has 475 pages. However, we still cannot determine the median number of pages for all the books on the two shelves. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

From statements one and two we know that there are a total of 49 books on the two shelves. Thus, we can determine which book, when the books are ordered by least number of pages to the greatest number of pages, is the median. To determine the median book we can follow the rule:

If a set of numbers is in numerical order and has n terms and if n is odd, the median is the value at the (n+1)/2 position.

position of median = (49 + 1)/2

position of median = 25

Thus, the 25th book is the median.

From our two statements we also know that the book with the greatest number of pages on the upper shelf has 400 pages, and the book with the least number of pages on the lower shelf has 475 pages. This means that all the books on the upper shelf have fewer pages than the books on the lower shelf. So if we ordered the books on both shelves from the least number of pages to the greatest number of pages, we see that the book on the top shelf, with the greatest number of pages, or the 25th book, would be the median. Since we know that the 25th book has 400 pages, we know the median is 400 pages.

If this is hard to see we also could create a diagram.

Answer: C

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