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 reaming page is 195. The torn page contains which of the following numbers?

Correct Option (B)

Let the booklet have $n$ pages. Pages are numbered starting from 1. The total sum of all page numbers is given by the formula: Sum$=\frac{n(n+1)}{2}$.

The sum of the numbers on the remaining pages is 195.

To determine the total number of pages ($n$) in the booklet, we identify the smallest $n$ for which the total sum of page numbers is greater than 195, as one page was torn.

• If \(n = 19\), the total sum would be \(\frac{19 \times 20}{2} = 190\). This sum is less than 195, which is not feasible if 195 is the sum of the remaining pages after a page was torn.
• If \(n = 20\), the total sum would be \(\frac{20 \times 21}{2} = 210\). This sum is greater than 195, indicating that 20 is the correct total number of pages in the booklet.

The original calculation demonstrates: $\frac{20\times 21}{2}=210=195+15$.

This implies that the sum of the numbers on the torn page is 15.

Since a torn page always contains two consecutive page numbers, we must identify two consecutive integers that sum to 15. These numbers are 7 and 8 ($7+8=15$).

Therefore, the torn page contains the numbers 7 and 8.

Incorrect Options:

• Option (A) 5, 6: The sum of these numbers is \(5 + 6 = 11\). This does not match the calculated sum of 15 for the torn page.

• Option (C) 9, 10: The sum of these numbers is \(9 + 10 = 19\). This does not match the calculated sum of 15 for the torn page.

• Option (D) 11, 12: The sum of these numbers is \(11 + 12 = 23\). This does not match the calculated sum of 15 for the torn page.