A cricketer has an average of 99 runs over 99 innings. How many runs must he score in his 100th innings so that his overall batting average becomes exactly 100 runs after 100 innings?

Introduction / Context:
This question uses the basic concept of average runs in cricket. It checks whether you can translate the verbal condition about the current average and the desired average into equations involving totals and then work backwards to find the required score in the next innings. Such problems help in strengthening the link between arithmetic mean and cumulative totals over time.

Given Data / Assumptions:

• Number of innings already played = 99.
• Current average runs = 99 runs per innings.
• Target average after 100 innings = 100 runs per innings.
• We assume all previous scores are fixed and only the 100th innings can change.

Concept / Approach:
The average is defined as total runs divided by number of innings. To reach a new average, we first compute the total runs required for that new average over 100 innings. We also compute the current total runs from the existing average over 99 innings. The difference between the required total and the current total is exactly the number of runs the batsman needs to score in the 100th innings.

Step-by-Step Solution:
Step 1: Current total runs = 99 innings * 99 runs per innings = 99 * 99 = 9801.Step 2: To average 100 runs over 100 innings, total required runs = 100 * 100 = 10000.Step 3: Runs needed in the 100th innings = 10000 - 9801.Step 4: 10000 - 9801 = 199.Step 5: Therefore, the batsman must score 199 runs in his 100th innings.

Verification / Alternative check:
If he scores 199 runs, the new total becomes 9801 + 199 = 10000. Dividing 10000 by 100 innings gives an average of exactly 100 runs, which matches the target average. This confirms that our calculation is accurate and that no arithmetic step has been missed.

Why Other Options Are Wrong:
Option 100 would lead to a total of 9901 runs, giving an average of 99.01, which is still below 100. Option 99 would keep the average almost unchanged. Option 101 would only give a total of 9902 runs. Option 150 gives a total of 9951 runs, which results in an average of 99.51, still less than 100.

Common Pitfalls:

• Thinking that an increase of 1 in the average requires only 1 extra run, without considering the number of innings.
• Forgetting to convert averages into totals before doing any subtraction.
• Making mistakes in squaring 99 or multiplying 100 by 100.

Final Answer:
The batsman must score 199 runs in his 100th innings to raise his average to 100.