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

Introduction / Context:
This cricket based aptitude question involves averages and how they change when a new performance is added. A batsman has some unknown average for his first 15 matches. After scoring 81 runs in his 16th match, his average increases by 3 runs. You must determine his new average after this 16th match. This is a common pattern in sports related average problems in competitive exams.

Given Data / Assumptions:
- The batsman has already played 15 matches before the 16th match.
- Let his average runs per match before the 16th match be a runs.
- Total runs in 15 matches before the 16th match = 15a.
- In the 16th match he scores 81 runs.
- As a result, his average increases by 3 runs, so new average after 16 matches is a + 3.
- We must find his average after the 16th match in runs per match.

Concept / Approach:
The idea is to translate the given information into an equation using the relationship average = total runs / number of matches. We express the new average after the 16th match in terms of the old total and the new score. Then we set this equal to the increased average a + 3. Solving this equation gives us the old average a, and therefore the new average a + 3.

Step-by-Step Solution:
Step 1: Let a be the batsman average runs per match for the first 15 matches.Step 2: Total runs in the first 15 matches = 15a.Step 3: In the 16th match he scores 81 runs, so new total runs after 16 matches = 15a + 81.Step 4: New average after 16 matches = (15a + 81) / 16.Step 5: We are told that this new average is 3 runs more than the old average a, so (15a + 81) / 16 = a + 3.Step 6: Multiply both sides by 16: 15a + 81 = 16a + 48.Step 7: Rearrange: 15a + 81 - 16a - 48 = 0 which simplifies to -a + 33 = 0.Step 8: Therefore a = 33.Step 9: New average after the 16th match = a + 3 = 33 + 3 = 36.

Verification / Alternative check:
If the old average is 33, the batsman total after 15 matches is 15 * 33 = 495 runs. After scoring 81 runs in the 16th match, the new total is 495 + 81 = 576 runs. The new average is 576 / 16 = 36 runs per match. The increase in average is 36 - 33 = 3 runs, which matches the condition in the question. This confirms that the new average is correct.

Why Other Options Are Wrong:
Values like 35, 34 or 33 do not produce the required 3 run increase when you back calculate the old average and total runs. Only an average of 36 is consistent with a score of 81 in the 16th match and the specified increase in average. You can verify by plugging each option back into the total and average formulas and checking whether the increase matches 3 runs.

Common Pitfalls:
Some learners mistakenly set (15a + 81) / 16 equal to 3 instead of a + 3, or they forget that the new average applies to 16 matches, not 15. Another common mistake is trying to guess the answer by trial without forming the equation, which can be slow and error prone. The clean algebraic method is fast, accurate and easy to reuse in similar problems.

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