On a shelf, two books, each 250 pages thick, are standing tightly next to each other (see figure). Each cover is 10 times thicker than the paper on which both books are printed. A bookmark is placed in each book.

The distance between the bookmarks is three times less than the total thickness of the two books. Between which pages is the bookmark in the second book, if in the first book it lies in the middle?

![](https://cdn.mathpix.com/cropped/2024_05_06_a16cba6403f5db62d5b1g-09.jpg?height=520&width=400&top_left_y=1661&top_left_x=838)
We will replace the cover of each book with ten sheets. Then, in total, the two books will have $250 \cdot 2 + 10 \cdot 4 = 540$ sheets. We can reason in different ways.

The first method. The distance between the bookmarks is $540 : 3 = 180$ sheets. In the first book, after the bookmark, there are $250 : 2 + 10 = 135$ sheets. Then, in the second book, before the bookmark: $180 - 135 = 45$ sheets, of which 10 sheets are the cover. Therefore, the bookmark lies after the sheet numbered $45 - 10 = 35$, that is, between sheets 35 and 36.

The second method. The total thickness of the two books is equal to the thickness of $250 \cdot 2 + 10 \cdot 4 = 540$ sheets. Therefore, the thickness of each cover is $1 / 54$ of the total thickness of the two books. Between the bookmarks is one-third of the two books, of which one-quarter is half of the first book, and $1 / 54$ is the cover of the second book. Thus, the number of sheets in the second book from the cover to the bookmark is $1 / 3 - 1 / 4 - 1 / 54 = 7 / 108$ of the total thickness of both books. It equals $540 \cdot 7 / 108 = 35$, that is, the bookmark lies between sheets 35 and 36.

## Answer

Between sheets 35 and 36.