A batsman scores 81 runs in his 16th match and thereby increases his average runs per match by 3 runs. What is his average (in runs per match) after the 16th match?

Introduction / Context:
This problem involves the concept of batting average and how it changes when a new performance is added. It tests your ability to form an equation using the relationship between total runs, number of matches, and average, and then solve for the unknown average after the new match.

Given Data / Assumptions:
- Number of matches before the 16th match = 15. - Let the old average runs per match be a runs. - Runs scored in the 16th match = 81. - After the 16th match, the average increases by 3 runs, so new average = a + 3. - We need the new average after 16 matches.

Concept / Approach:
Average runs per match = total runs scored divided by number of matches. Before the 16th match, total runs are 15 * a. After the 16th match, total runs are 15 * a + 81 and number of matches is 16. We are told that this new total divided by 16 equals a + 3. This leads to a linear equation in a, which we can solve easily and then add 3 to get the new average.

Step-by-Step Solution:
Step 1: Let old average after 15 matches be a runs per match. Step 2: Total runs after 15 matches = 15 * a. Step 3: In the 16th match, he scores 81 runs. Step 4: Total runs after 16 matches = 15 * a + 81. Step 5: New average after 16 matches is a + 3 runs per match. Step 6: Set up the equation using the definition of average: (15 * a + 81) / 16 = a + 3. Step 7: Multiply both sides by 16: 15 * a + 81 = 16 * (a + 3). Step 8: Expand the right side: 16 * a + 48. Step 9: So 15 * a + 81 = 16 * a + 48. Step 10: Rearrange terms: 81 - 48 = 16a - 15a. Step 11: 33 = a, so old average = 33 runs per match. Step 12: New average after the 16th match = a + 3 = 33 + 3 = 36 runs per match.

Verification / Alternative check:
We can verify by computing totals directly. If the old average was 33, then total runs after 15 matches = 15 * 33 = 495. After scoring 81 in the 16th match, total runs = 495 + 81 = 576. New average after 16 matches = 576 / 16 = 36 runs per match. The increase in average is 36 - 33 = 3 runs, which matches the problem statement. This confirms that the calculations are correct.

Why Other Options Are Wrong:
Option A (35): Would give an old average of 32 runs and does not match the 81 runs condition correctly. Option B (34): Leads to an inconsistent total when checked with the given increase. Option C (33): This is the old average, not the new average after the improvement. Option E (32): Too low and inconsistent with the increase of 3 runs after such a high score of 81 in the latest match.

Common Pitfalls:
Many students mistakenly treat 81 runs as the new average or confuse total runs with average runs. Another error is to forget that the number of matches changes from 15 to 16 and to use 15 in the denominator for the new average. It is crucial to write a clear equation based on the definition of average and carefully track the number of matches before and after the new performance.

Final Answer:
The batsman's average after the 16th match is 36 runs per match.