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 sum of page numbers from 1 to N is given by the formula N(N+1)/2. The sum of the remaining pages is 195. This means the original total sum of pages must be slightly greater than 195. Let's estimate the total number of pages (N): If N=19, the sum is 19 * (19+1) / 2 = 19 * 20 / 2 = 190. This is less than 195, so N must be greater than 19. If N=20, the sum is 20 * (20+1) / 2 = 20 * 21 / 2 = 210. This is a plausible total sum. If N=21, the sum is 21 * (21+1) / 2 = 21 * 22 / 2 = 231. Let's assume the booklet had 20 pages. The total sum of page numbers would be 210. The sum of the torn page numbers = Total sum - Sum of remaining pages Sum of torn page numbers = 210 - 195 = 15. A page always contains two consecutive numbers. Let these numbers be P and P+1. Their sum is P + (P+1) = 2P + 1. So, 2P + 1 = 15. 2P = 14. P = 7. The torn page numbers are 7 and 8. If we had assumed N=21, the sum of torn page numbers would be 231 - 195 = 36. Then 2P + 1 = 36, which gives 2P = 35, and P = 17.5. Page numbers must be integers, so N cannot be 21. Thus, the booklet had 20 pages, and the torn page contained numbers 7 and 8. Analyzing the options: A) 5,6: Sum = 11. (Incorrect) B) 7,8: Sum = 15. (Correct) C) 9,10: Sum = 19. (Incorrect) D) 11,12: Sum = 23. (Incorrect) The final answer is B