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

Introduction / Context:
This is a typical cricket based averages problem. You are given that a batsman has some unknown average for a certain number of innings. In the next innings he scores a particular number of runs and his overall average increases by a specific amount. From this information, you must find the new average. This question tests your understanding of how an additional performance affects the overall average.

Given Data / Assumptions:
- Number of innings before the latest innings = 12.
- Let the old average runs per innings be a runs.
- In the 13th innings, the batsman scores 96 runs.
- After this innings, his average increases by 5 runs, so the new average is a + 5.
- We need the new average after 13 innings.

Concept / Approach:
The total runs before the 13th innings can be expressed as 12a. After the 13th innings, the total becomes 12a + 96. The new average is this total divided by 13. We know that this new average is equal to a + 5. By setting up this equation and solving for a, we can find both the old and the new average. This approach is standard for average increment problems in cricket and other sports contexts.

Step-by-Step Solution:
Step 1: Old total runs in 12 innings = 12a.Step 2: After scoring 96 runs in the 13th innings, new total runs = 12a + 96.Step 3: New average after 13 innings = (12a + 96) / 13.Step 4: Given that this new average is a + 5, so (12a + 96) / 13 = a + 5.Step 5: Multiply both sides by 13: 12a + 96 = 13a + 65.Step 6: Rearrange: 12a + 96 - 13a - 65 = 0, which gives -a + 31 = 0.Step 7: Hence a = 31. This is the old average.Step 8: New average after 13 innings = a + 5 = 31 + 5 = 36.

Verification / Alternative check:
If the old average is 31, total runs after 12 innings are 12 * 31 = 372. Adding 96 gives new total runs of 468. The new average is 468 / 13 = 36 runs per innings. The increase in average is 36 - 31 = 5 runs, which matches the information given. So the calculation is consistent and confirms that the correct new average is 36.

Why Other Options Are Wrong:
If the new average were 32, 28, 40 or 42, the total runs computed from these averages would not match both the old and new average conditions. For example, a new average of 32 would imply too low a total, while 40 or 42 would imply too high a total compared to what is obtained by adding 96 runs to the old total.

Common Pitfalls:
Students sometimes incorrectly set (12a + 96) / 13 equal to 5 instead of a + 5, or they forget to use 13 as the divisor for the new average, still dividing by 12. Carefully track the number of innings and the definition of average to avoid these mistakes.

Final Answer:
The new batting average after 13 innings is 36 runs per innings.