A batsman scores 100 runs in the 25th match of his career, which increases his average runs per match by 1.4 runs. What was his average runs per match before playing this 25th match?

Introduction / Context:
This question again uses the concept of average in a sports setting. By knowing the score in a particular match and the change in the overall average, we can work backwards to determine the previous average performance. This is a standard pattern in aptitude tests and helps strengthen understanding of averages and algebraic equations.

Given Data / Assumptions:

• Before the 25th match, the batsman had already played 24 matches.
• Runs scored in the 25th match = 100.
• Average runs per match increases by 1.4 after this match.
• Let the old average before 25th match be a runs per match.

Concept / Approach:
Average is total runs divided by number of matches. Before the 25th match, the total runs are 24 * a. After scoring 100 runs, total runs become 24 * a + 100 and the number of matches becomes 25. This new average is given to be a + 1.4. We form an equation based on this relationship and solve for a.

Step-by-Step Solution:
Let old average = a runs per match. Total runs before 25th match = 24 * a. After 25th match, total runs = 24 * a + 100. Number of matches after 25th match = 25. New average = (24 * a + 100) / 25. Given new average = a + 1.4. So, (24 * a + 100) / 25 = a + 1.4. Multiply both sides by 25: 24 * a + 100 = 25 * a + 35. Rearrange terms: 24 * a + 100 - 35 = 25 * a. 24 * a + 65 = 25 * a. 65 = a. Therefore, old average = 65 runs per match.
Verification / Alternative check:
Total runs before 25th match = 24 * 65 = 1560. After scoring 100, total runs = 1560 + 100 = 1660. New average = 1660 / 25 = 66.4 runs per match. Increase in average = 66.4 - 65 = 1.4 runs, which matches the given condition.
Why Other Options Are Wrong:
Option B (55): Leads to a different new average that does not match the given increase. Option C (75): Produces an incorrect change in average when used in the equation. Option D (45): Far from the actual value and does not satisfy the relation. Option A (65): Correct, since it satisfies the equation and the given increase in average.
Common Pitfalls:
Confusing the number of matches before and after the innings can lead to wrong equations. Some students attempt to simply average 100 with a guess for the previous average, which is incorrect. Others may forget that the increase in average applies to all matches, not just the latest one.
Final Answer:
The batsman's average runs per match before the 25th match was 65 runs.