IAS 2020 — CSAT Maths

Correct Option (B)

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

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

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

• If $n = 19$n=19, the total sum would be $\frac{19 \times 20}{2} = 190$219×20​=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 $n = 20$n=20, the total sum would be $\frac{20 \times 21}{2} = 210$220×21​=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$220×21​=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$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$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$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$11+12=23. This does not match the calculated sum of 15 for the torn page.