The average of the runs scored by a cricketer in his first 99 innings is 99. How many runs must he score in his 100th innings so that his average over 100 innings becomes exactly 100?

Introduction / Context:
This is a classic cricket and averages question that appears frequently in aptitude exams. It deals with how a single new observation (the 100th innings) affects the overall average after many innings. The problem highlights that even a large additional score may change the average by only a small amount when many prior innings are already included.

Given Data / Assumptions:

Number of innings already played = 99.
Average runs over these 99 innings = 99 runs per innings.
We want the average over 100 innings to be 100 runs per innings.
Runs in the 100th innings are unknown and need to be found.

Concept / Approach:
Average runs = total runs / number of innings. We first find the total runs scored in the first 99 innings using the existing average. Then we compute the total runs required after 100 innings to have an average of 100. The difference between the required total and the existing total gives the runs needed in the 100th innings. This is a straightforward but important application of average and total relationships.

Step-by-Step Solution:
Step 1: Compute total runs in the first 99 innings. Total runs after 99 innings = 99 * 99 = 9801. Step 2: Compute the total runs required for an average of 100 after 100 innings. Required total after 100 innings = 100 * 100 = 10000. Step 3: Find the runs needed in the 100th innings. Runs in 100th innings = required total - current total. Runs needed = 10000 - 9801 = 199.

Verification / Alternative Check:
If the batsman scores 199 runs in his 100th innings, then his total runs become 9801 + 199 = 10000. Dividing by 100 innings gives 10000 / 100 = 100 runs per innings, which matches the required new average. This confirms that 199 is indeed the correct score needed.

Why Other Options Are Wrong:
99 or 100 runs would give totals of 9900 or 9901 respectively, leading to averages less than 100.
101 runs would give a total of 9902, yielding an average of only 99.02, still below 100.

Common Pitfalls:
Some students make the mistake of thinking that adding 1 run to the average means scoring only a little more in the next innings, such as 101 instead of 99. They forget that the new innings must pull up the average of all 100 innings, not just itself. The correct approach is always to work through total runs: multiply the average by the number of innings to get totals, then find the difference required to reach the new total.

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