In an examination, a student's average mark is 63 per paper. If he had obtained 20 more marks in Geography and 2 more marks in History, his average mark per paper would have become 65. How many papers did he take in the examination?

Introduction / Context:
This average-based problem involves a student who has a certain average mark per paper. When we hypothetically add extra marks to two specific subjects (Geography and History), his average would increase. The key idea is to understand how changes in total marks relate to changes in average marks, allowing us to find the number of papers in the exam.

Given Data / Assumptions:

• Original average mark = 63 marks per paper.
• He took n papers (we need to find n).
• If he had scored 20 more marks in Geography and 2 more marks in History, his average would become 65 marks per paper.
• Total additional marks in that hypothetical scenario = 20 + 2 = 22 marks.

Concept / Approach:
Let n be the number of papers. The original total marks are 63 * n. If we increase his marks by 22 in total, the new total would be 63 * n + 22. We are told that this new total, when divided by n, would give a new average of 65 marks. This gives one linear equation in n, which we can solve easily.

Step-by-Step Solution:
Step 1: Let the number of papers be n. Step 2: Original total marks = 63 * n. Step 3: Additional marks (20 in Geography + 2 in History) = 22 marks. Step 4: Hypothetical new total marks = 63 * n + 22. Step 5: New average is given to be 65 marks per paper. Therefore, (63 * n + 22) / n = 65. Step 6: Multiply both sides by n: 63n + 22 = 65n. Step 7: Rearrange: 22 = 65n - 63n = 2n. Step 8: So n = 22 / 2 = 11.
Verification / Alternative check:
If n = 11, original total marks = 63 * 11 = 693. Hypothetical new total = 693 + 22 = 715. New average = 715 / 11 = 65 marks per paper, which matches the condition.
Why Other Options Are Wrong:
If n were 8, 9, 10, or 12, the equation (63n + 22) / n = 65 would not hold. For example, if n = 10, new average = (630 + 22) / 10 = 652 / 10 = 65.2, not exactly 65.
Common Pitfalls:
Some students mistakenly divide the extra 22 marks directly by the difference in averages (2) and stop, not realizing that this effectively already assumes the correct approach; however, they may misinterpret the result as additional papers instead of the total number of papers. Others forget that the average change applies across all papers, not just the two with extra marks.
Final Answer:
The student appeared in 11 papers in the examination.