We are given that each of the 25 books on bookshelf X is thinner than each of the 25 books on bookshelf Y.
⇒ (Thickness of the thickest book on bookshelf X) < (Thickness of the thinnest book on bookshelf Y)
You must not assume that the thickness of each of the 25 books on bookshelf X is the same; they all may be of different thickness; they all may be of the same thickness or a few of them may be of the same thickness. The same goes for the books on bookshelf Y.
We are asked to determine whether the median thickness of the 50 books is less than 20 millimeters.
You must know that to find out the median of a dataset, the elements of the dataset must be arranged in an ascending or in a descending order.
Let's assume that that all the 50 books are arranged as per the ascending order of their thickness. All 25 books of bookshelf X would be on the left-hand side, arranged as per the increasing order of the thickness of books, and all 25 books of bookshelf Y would be on the right-hand side, arranged as per the increasing order of the thickness of books. Thus, in this ordered arrangement of the 50 books, the thickest book of bookshelf X (25th book in order) would be immediately left of the thinnest book of bookshelf Y (26th in order).
You must know that:
(Median value of a dataset) = (Value of the middlemost element)
When the number of elements in the dataset is even, the median is equal to the average of the values of the two middlemost elements.
(the median thickness of 50 books)=(Average of the thickness of 25th book and the thickness of 26th book in the ordered arrangement)
We know that the 26th book is the 1st book on bookshelf Y.
So, to get the value of median thickness of 50 books, we must know that value of the thickness of the thickest book of bookshelf X and of the thinnest book of bookshelf Y.
Note: In the above analysis, we arranged the books on the two shelves in ascending order of their thickness. If you chose to arrange them in descending order of thickness instead, that is fine too. The analysis remains the same.
The thickness of the thinnest book on shelf X = 2 millimeters
This information is of no use to us as we want the thickness of the 25th and the 26th book to calculate the median. - Insufficient
(Thickness of the thinnest book on shelf Y) = (Thickness of the 26th book in ascending order of thickness) = 20-millimeters
Though we do not know the thickness of the 25th book, we can make a deduction about the median. The question does not ask the value of the median; it asks whether the median thickness of the 50 books is less than 20 millimeters. This is a "Yes/No" type of question.
Since (the thickness of 26th book) = 20 millimeters, and we're given that even the thickest book on bookshelf X is thinner than the thinnest book on bookshelf Y, we can write: (the thickness of 25th book) < 20 millimeters.
Say (the thickness of 25th book)=19.99 millimeters. Thus, Median
=19.99 + 20 / 2 < 20 millimeters. The answer is Yes. - Sufficient