A batsman has a certain average of runs for 12 innings. In the 13th inning he scores 96 runs, and his average runs per inning increases by 5 runs. What will be his new batting average after the 13th inning?

Introduction / Context:
This problem is about averages in the context of cricket scores. When a batsman plays another inning and scores runs significantly higher than his current average, his average increases. By relating the original and new totals through the change in average, we can find the new average directly.

Given Data / Assumptions:
• The batsman has played 12 innings initially.• His average after these 12 innings is some unknown value A.• In the 13th inning he scores 96 runs.• After the 13th inning, his average increases by 5 runs, becoming A + 5.• We need to find the value of A + 5, the new average.

Concept / Approach:
The total runs scored in 12 innings is 12A. After scoring 96 in the 13th inning, the total becomes 12A + 96. This new total divided by 13 equals the new average A + 5. This gives an equation in A which can be solved algebraically. Once A is known, A + 5 gives the required average after 13 innings.

Step-by-Step Solution:
Step 1: Let the original average after 12 innings be A runs.Step 2: Total runs scored in 12 innings = 12A.Step 3: In the 13th inning he scores 96 runs, so new total runs = 12A + 96.Step 4: New average after 13 innings is given as A + 5.Step 5: Therefore, (12A + 96) / 13 = A + 5.Step 6: Multiply both sides by 13: 12A + 96 = 13(A + 5).Step 7: Expand right side: 13(A + 5) = 13A + 65.Step 8: So 12A + 96 = 13A + 65.Step 9: Rearranging, 96 - 65 = 13A - 12A gives 31 = A.Step 10: New average = A + 5 = 31 + 5 = 36.

Verification / Alternative check:
With A = 31, total after 12 innings is 12 * 31 = 372. Adding the 13th inning score, total runs = 372 + 96 = 468. Average after 13 innings = 468 / 13 = 36, which is indeed 5 more than 31. So the result is consistent with the problem statement.

Why Other Options Are Wrong:
28, 32, 42: These do not produce a 5 run increase from a consistent original average when 96 is added as the 13th inning score. Substituting them back into the equation leads to contradictions.

Common Pitfalls:
Students may incorrectly assume the original average or forget that averages depend on total runs and total innings. Another common error is to equate the 13th inning score directly with the increase in average without forming the algebraic equation. Using the proper relation between old total, new total, and new average avoids these issues.

Final Answer:
The batsman's average after the 13th inning is 36 runs per inning.