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

Introduction / Context:
This question is another example of how averages change when a new data point is added, here in the context of cricket scores. It checks your skill in forming an equation from the given information and then solving for the new batting average.

Given Data / Assumptions:
- Number of matches before the 10th match = 9. - Let the old average runs per match be a runs. - Runs scored in the 10th match = 111 runs. - After the 10th match, the average increases by 5 runs, so new average = a + 5. - We need the new average after 10 matches.

Concept / Approach:
Before the 10th match, total runs = 9 * a. After the 10th match, total runs become 9 * a + 111 and the number of matches becomes 10. The new average is given as a + 5. So we can write:
(9 * a + 111) / 10 = a + 5. This linear equation in a can be solved to find the old average, and then we add 5 to get the new average.

Step-by-Step Solution:
Step 1: Let old average runs per match be a. Step 2: Total runs after 9 matches = 9 * a. Step 3: In the 10th match, the batsman scores 111 runs. Step 4: Total runs after 10 matches = 9 * a + 111. Step 5: New average after 10 matches = a + 5. Step 6: Form the equation using the definition of average: (9 * a + 111) / 10 = a + 5. Step 7: Multiply both sides by 10: 9 * a + 111 = 10 * (a + 5). Step 8: Expand the right side: 10 * a + 50. Step 9: So 9 * a + 111 = 10 * a + 50. Step 10: Rearrange terms: 111 - 50 = 10a - 9a. Step 11: 61 = a, so old average = 61 runs per match. Step 12: New average after the 10th match = a + 5 = 61 + 5 = 66 runs per match.

Verification / Alternative check:
Check with direct totals. If old average = 61, total after 9 matches = 9 * 61 = 549. After scoring 111 runs in the 10th match, total runs = 549 + 111 = 660. New average = 660 / 10 = 66 runs per match. Increase in average is 66 - 61 = 5 runs, which matches the question statement perfectly. This verifies the result.

Why Other Options Are Wrong:
Option B (61): This is the old average, not the new average. Option C (62) and D (64): Do not satisfy the equation when checked back against the 111 run performance. Option E (58): Too low and inconsistent; it would suggest a decrease rather than an increase in average.

Common Pitfalls:
One common mistake is to directly average 111 with the old average or to forget that the number of matches changes from 9 to 10. Another mistake is to set (9 * a + 111) / 9 instead of dividing by 10 for the new average. Always ensure that the denominator equals the new number of matches and that the increase in average is applied to the old average, not to the new score itself.

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