A batsman scores 95 runs in his 13th match and thereby increases his average runs per match by 4. What is his batting average (in runs per match) after the 13th match?

Introduction / Context:
This is a typical cricket average problem. We are given how a player's average changes after one additional match and the score in that match. Using this information, we can find both the original average and the new average after the 13th match.

Given Data / Assumptions:

• Number of matches before the latest innings = 12.
• The latest (13th) innings score = 95 runs.
• After this 13th innings, the batsman's average increases by 4 runs per match.
• We need to find the new average after 13 innings.

Concept / Approach:
Let the old average after 12 innings be a runs. Then:

• Total runs in first 12 innings = 12 * a.
• After the 13th innings, total runs = 12 * a + 95.
• New average after 13 innings = a + 4 (since it increases by 4).

We use the definition of average: New average = (total runs after 13 innings) / 13. This gives an equation in a that we can solve.

Step-by-Step Solution:
Step 1: Let old average after 12 innings be a runs. Total runs after 12 innings = 12 * a. Step 2: Express total runs after the 13th innings. Total after 13 innings = 12 * a + 95. Step 3: Express new average in terms of a. New average = a + 4. By definition, new average = (12 * a + 95) / 13. Step 4: Form the equation and solve. (12 * a + 95) / 13 = a + 4. 12 * a + 95 = 13 * (a + 4) = 13 * a + 52. 12 * a + 95 = 13 * a + 52. 95 - 52 = 13 * a - 12 * a. 43 = a. Step 5: Compute the new average. New average = a + 4 = 43 + 4 = 47.

Verification / Alternative check:
If the old average was 43, total runs in 12 innings were 12 * 43 = 516. After scoring 95, total runs become 516 + 95 = 611. Now, 611 / 13 = 47 runs per innings, which is 4 more than 43. This matches the condition that the average increased by 4, confirming the result.

Why Other Options Are Wrong:
If the new average were 43, then the average would not have increased. If it were 45 or 49, solving backward would not give an integer old average and would not match the given 95 runs. Only 47 satisfies the equation relating old average, new average, and the additional score.

Common Pitfalls:
A common mistake is to treat 95 as the new average instead of a score, or to simply add 4 to 95. Another error is mixing up total runs and average. Always start by defining the original average as a variable, computing total runs, and then forming the equation with the new average after including the latest score.

Final Answer:
The batsman's average after the 13th match is 47 runs per match.