UPSC Prelims 2020 CSAT Paper II · Q14 of 80 Basic Numeracy medium

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?

  1. (a) 5, 6
  2. (b) 7, 8 ✓ UPSC's answer
  3. (c) 9, 10
  4. (d) 11, 12

Why the answer is (b)

• A torn page in a booklet consists of two consecutive numbers, where the first number is odd and the second is even (e.g., 1-2, 3-4, 5-6, etc.).

• Let the total number of pages be $n$. The sum of all page numbers is given by the formula $\frac{n(n+1)}{2}$.

• The sum of the remaining pages is 195, so the sum of the torn page numbers must be $\frac{n(n+1)}{2} - 195$.

• Since the sum of two consecutive integers (odd + even) is always odd, the total sum $\frac{n(n+1)}{2}$ must be even (because Even - Odd = Odd is false; actually, Total Sum = Remaining Sum + Torn Sum. 195 is odd. Torn sum is odd. Odd + Odd = Even. So the total sum must be even).

• Testing small values of $n$: If $n=20$, total sum is $\frac{20 \times 21}{2} = 210$. The torn sum is $210 - 195 = 15$.

• The pair of consecutive numbers summing to 15 is 7 and 8 (since $7+8=15$). This matches option (b).

Why the other options are wrong

(a) 5, 6
The sum of 5 and 6 is 11, which would imply a total sum of 206, not a triangular number.
(c) 9, 10
The sum of 9 and 10 is 19, which would imply a total sum of 214, not a triangular number.
(d) 11, 12
The sum of 11 and 12 is 23, which would imply a total sum of 218, not a triangular number.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.

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