One page is torn from a booklet whose pages are numbered in the usual manner starting from the first page as 1. The sum of the numbers on the remaining pages is 195. The torn page contains which of the following numbers?

The correct option is 7, 8.

Let us solve this step-by-step to find which page was torn from the booklet.

Step 1: Understand the booklet's page numbering system
In any standard booklet, page numbering starts at page 1. When a single sheet (one page) is torn out of a booklet, it removes two consecutive page numbers: an odd page number on the front and the next even page number on the back (or vice-versa, i.e., numbers of the form $x$ and $x+1$).

Step 2: Set up the sum formula
Let $n$ be the total number of pages originally in the booklet.
The sum of the first $n$ natural numbers is given by the formula:

$S=\frac{n(n+1)}{2}$

Let the two page numbers on the torn sheet be $x$ and $x+1$.
The sum of the remaining pages is given as 195. Therefore:

$\frac{n(n+1)}{2}-(x+x+1)=195$

$\frac{n(n+1)}{2}-(2x+1)=195$

Step 3: Determine the total number of pages ($n$)
Since $2x+1>0$, the total sum $S$ must be strictly greater than 195.
Let's find the smallest integer $n$ such that $n(n+1)2>195$:

If $n=19$:
$S=\frac{19\×20}{2}=190$ (This is less than 195, which is impossible).

If $n=20$:
$S=\frac{20\×21}{2}=210$ (This is greater than 195, so $n=20$ is a valid candidate).

Step 4: Calculate the torn page numbers
Using $n=20$, the original sum of all pages is 210.
The sum of the two torn page numbers is:

$210-195=15$

Since the torn page numbers are consecutive, let them be $x$ and $x+1$:

$x+(x+1)=15$

$2x+1=15$

$2x=14$

$x=7$

Thus, the numbers on the torn page are 7 and 8.