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

Introduction / Context:
This question is again about cricket averages and the effect of one innings on a batsman overall average. The batsman scores 100 runs in his 25th match, and this raises his average by 1.4 runs per match. From this information, we need to determine his average before this match. This is a standard type of average increment problem that appears in competitive exams.

Given Data / Assumptions:
- Before the 25th match, the batsman has played 24 matches.
- Let his old average be a runs per match.
- In the 25th match he scores 100 runs.
- New average after 25 matches is a + 1.4 runs per match.
- We must calculate the old average a.

Concept / Approach:
We compute the total runs before the 25th match as 24a. After the 25th match, the total runs become 24a + 100. The new average is this new total divided by 25 and is given to be a + 1.4. Setting up the equation (24a + 100) / 25 = a + 1.4 allows us to solve for a using simple algebra. This method is common for problems where a new data point changes the overall average.

Step-by-Step Solution:
Step 1: Old total runs in 24 matches = 24a.Step 2: After the 25th match, new total runs = 24a + 100.Step 3: New average after 25 matches = (24a + 100) / 25.Step 4: This new average is given to be a + 1.4, so (24a + 100) / 25 = a + 1.4.Step 5: Multiply both sides by 25: 24a + 100 = 25a + 35.Step 6: Rearrange: 24a + 100 - 25a - 35 = 0, which gives -a + 65 = 0.Step 7: Hence a = 65.Step 8: Therefore, the batsman average before the 25th match was 65 runs per match.

Verification / Alternative check:
If a = 65, then total runs after 24 matches are 24 * 65 = 1560 runs. After scoring 100 runs in the 25th match, new total runs = 1560 + 100 = 1660. New average = 1660 / 25 = 66.4 runs per match. The increase in average is 66.4 - 65 = 1.4 runs, which confirms that the old average of 65 is correct.

Why Other Options Are Wrong:
If the batsman original average were 55, 75, 45 or 70, the algebraic equation derived from the given condition would not hold, and the new average would not increase by exactly 1.4 runs after scoring 100. Only the value 65 satisfies both the relationship between old and new totals and the given increment in average.

Common Pitfalls:
Common errors include using 24 instead of 25 in the denominator for the new average or forgetting to add 100 to the old total when forming the new total. Some students also mistakenly assume that 100 divided by 25 is the increase in average without considering previous totals. Always form the equation carefully from total runs and the definition of average.

Final Answer:
The batsman average before the 25th match was 65 runs per match.