A batsman scores 111 runs in his 10th match and thereby increases his average runs per match by 5 runs. What is his batting average after the 10th match?

Introduction / Context:
This is another cricket average question involving a change in average after a new innings. The batsman has an unknown average over his first 9 matches. After scoring 111 runs in the 10th match, his average increases by 5 runs. You must determine his new average after the 10th match. This question reinforces the same idea of average and total runs used in many sports and exam scenarios.

Given Data / Assumptions:
- The batsman has played 9 matches before the 10th match.
- Let his average runs per match before the 10th match be a runs.
- Total runs in 9 matches before the 10th match = 9a.
- In the 10th match he scores 111 runs.
- His average increases by 5 runs, so new average after 10 matches is a + 5.
- We are asked to find this new average after 10 matches.

Concept / Approach:
As before, average is total runs divided by number of matches. Write the expression for the new average in terms of the old total and the new score, and set it equal to a + 5. Solving this simple linear equation provides the old average a, and adding 5 gives the required new average. This clear algebraic approach is the standard way to tackle such problems.

Step-by-Step Solution:
Step 1: Let the old average across 9 matches be a runs per match.Step 2: Total runs in the first 9 matches = 9a.Step 3: In the 10th match he scores 111 runs, so the new total runs after 10 matches = 9a + 111.Step 4: New average after 10 matches = (9a + 111) / 10.Step 5: The problem tells us that this new average is 5 runs more than the old average, so (9a + 111) / 10 = a + 5.Step 6: Multiply both sides by 10: 9a + 111 = 10a + 50.Step 7: Rearrange: 9a + 111 - 10a - 50 = 0 which simplifies to -a + 61 = 0.Step 8: Therefore a = 61.Step 9: New average after the 10th match = a + 5 = 61 + 5 = 66 runs per match.

Verification / Alternative check:
If the old average was 61, the batsman total after 9 matches was 9 * 61 = 549 runs. After scoring 111 runs in the 10th match, his new total becomes 549 + 111 = 660 runs. Dividing by 10 matches gives a new average of 660 / 10 = 66 runs per match. The increase in average is 66 - 61 = 5 runs, exactly as the question describes. This verifies the correctness of the calculation.

Why Other Options Are Wrong:
A new average of 61, 62, 64 or 60 does not satisfy the condition of a 5 run increase after scoring 111. For each of these, back calculating the old average and checking total runs leads to contradictions. Only the value 66 produces an old average of 61 and the correct total of 660 runs after 10 matches.

Common Pitfalls:
Common mistakes include using 9 instead of 10 in the denominator when computing the new average or thinking that the new average is 111 / 10. Another error is to confuse the increase in total runs with the increase in average. Always keep track of how many matches the average applies to, and build your equation from totals and averages carefully.

Final Answer:
The batsman average after the 10th match is 66 runs per match.