A cricket player has an average of 32 runs per innings over his first 10 innings. How many runs must he score in his 11th innings so that his overall batting average increases by 4 runs and becomes 36 runs per innings?

Introduction / Context:
This problem tests the concept of arithmetic mean (average) and how it changes when additional data is added. We use the relationship between total runs, number of innings, and average score to determine how many runs the batsman must score in his next innings to raise his average by a fixed amount.

Given Data / Assumptions:

• Number of initial innings = 10.
• Average runs for these 10 innings = 32 runs per innings.
• We add one more innings (the 11th innings).
• Required new average after 11 innings = 32 + 4 = 36 runs per innings.
• Let the runs scored in the 11th innings be R.

Concept / Approach:
Average is defined as: average = (sum of observations) / (number of observations). We first compute the total runs scored in the first 10 innings using the given average. Then, we express the new total runs after including the 11th innings in terms of R. By equating this new total to the required average multiplied by 11 innings, we can solve for R.

Step-by-Step Solution:
Step 1: Total runs in first 10 innings = 32 * 10 = 320. Step 2: Let runs in 11th innings be R. Then new total runs = 320 + R. Step 3: Required new average = 36 runs over 11 innings. So, (320 + R) / 11 = 36. Step 4: Multiply both sides by 11: 320 + R = 36 * 11. Step 5: Compute 36 * 11 = 396. Step 6: So 320 + R = 396 → R = 396 - 320 = 76.
Verification / Alternative check:
Check the final average: total runs = 320 + 76 = 396. Average over 11 innings = 396 / 11 = 36 runs per innings, which matches the required value.
Why Other Options Are Wrong:
Option B (79), C (85), D (87) and E (92) all produce totals different from 396, therefore the resulting averages would not equal 36 when divided by 11 innings. For example, if R = 79, new total = 399 and average = 399 / 11 = 36.27 (approx), not exactly 36.
Common Pitfalls:
A common error is to add 4 directly to the previous total or to the previous average without accounting for the change in the number of innings. Another mistake is to forget that the new average applies to all 11 innings, not just the latest one.
Final Answer:
The batsman must score 76 runs in his next innings to increase his average from 32 to 36 runs per innings.