एक पुस्तिका, जिसके पन्ने सामान्य रूप से हैं, पहले पन्ने की संख्या 1 से प्रारंभ करते हुए अंकित हैं। इस पुस्तिका से एक पन्ना फाड़ लिया जाता है। बचे हुए पन्नों पर अंकित पृष्ठ संख्याओं का योग 195 है। फटे हुए पन्ने पर निम्न में से कौन-सी संख्याएँ हैं?
सही उत्तर: (c) 9, 10
पुस्तिका के पृष्ठों की कुल संख्या n का कुल योग n(n + 1) / 2 होता है। मान लेते हैं कि फाड़े गए पन्ने पर संख्याएँ k और k + 1 हैं। शेष योग 195 है, अतः कुल योग = 195 + k + (k + 1) = 196 + 2k। यदि कुल पृष्ठ 20 हों, तो कुल योग 210 होगा। 210 - 195 = 15, जो कि फटे पन्ने की संख्याओं का योग है। 7 और 8 का योग 15 है, तथा इनका योग समीकरण को संतुष्ट करता है। अतः फटे हुए पन्ने पर 7 और 8 हैं।
In English (Question & Model Answer)
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?
Correct Option: (c) 9, 10
Let the booklet have n pages. The sum of page numbers from 1 to n is n(n + 1) / 2. When one page containing two consecutive page numbers (say k and k + 1) is torn, the sum of the remaining page numbers is given as 195. Therefore, total sum S = n(n + 1) / 2 = 195 + k + (k + 1) = 196 + 2k. Let's test values of n to find a suitable sum. If n = 20, total sum = 20(21)/2 = 210. Then 210 - 195 = 15, which is the sum of the torn page numbers. Since the torn page has two consecutive numbers k and k + 1 whose sum is 15: k + k + 1 = 15 => 2k = 14 => k = 7. Wait! If k = 7, the page numbers are 7 and 8. Let's check total sum with n = 20: 210 - (7 + 8) = 210 - 15 = 195. This matches the given sum of 195! Wait, let's check the options: (a) 5, 6, (b) 7, 8, (c) 9, 10, (d) 11, 12. Since 7, 8 is option (b), let's verify if n=20 works. Wait, if pages are 1 to 20, the page numbers on a single physical leaf are always an even number on the left and an odd number on the right (e.g., page 8 and 9, or page 7 and 8? Wait! In a booklet, page 1 is right side, page 2 and 3 are back-to-back, etc. A torn page is a single physical leaf having page numbers k and k+1 where one is even and one is odd. If k = 7 and k+1 = 8, 7 is odd and 8 is even, which means page 7 and 8 are on the same leaf!). Let's check if sum 195 with n=19 or others works. If sum is 195, n(n+1)/2 must be slightly greater than 195. Since 19(20)/2 = 190 (too small), 20(21)/2 = 210. Difference is 210 - 195 = 15. The two consecutive numbers summing to 15 are 7 and 8. Therefore, the torn page contains 7 and 8. Wait, option (b) is 7, 8.