Step 1: Understanding the Question:
The topic of this question is Averages, specifically how the average changes when a new value is added to a data set. We are given the number of innings played and the impact of the most recent score on the overall average. The objective is to calculate the updated average. This requires setting up an algebraic balance between the total runs before and after the 12th inning. It's a fundamental arithmetic mean problem found in most quantitative aptitude sections.
Step 2: Key Formulas and approach:
The main formula is $\text{Average} = \text{Total Runs} / \text{Total Innings}$. This implies $\text{Total Runs} = \text{Average} \times \text{Innings}$.
Our approach is:
1. Let the original average for 11 innings be $x$.
2. Express the total runs after 11 innings as $11x$.
3. Add the 12th inning score (90) to this total.
4. Set up an equation where this new total divided by 12 equals the new average $(x + 5)$.
5. Solve for $x$ and then calculate $(x + 5)$.
Step 3: Detailed Explanation:
We start with the assumption that the average for the first 11 innings was $x$. This means he scored a total of $11 \times x$ runs.
In the 12th inning, he scored 90 runs. So, his total runs after 12 innings became $11x + 90$.
His new average is stated to be 5 more than the old average. So, New Average = $x + 5$.
Using the formula: $(\text{Total Runs after 12 innings}) / 12 = \text{New Average}$.
This gives the equation: $(11x + 90) / 12 = x + 5$.
Multiply both sides by 12 to clear the fraction: $11x + 90 = 12(x + 5)$.
Expanding the brackets: $11x + 90 = 12x + 60$.
Rearranging the terms to solve for $x$: $90 - 60 = 12x - 11x$, which means $30 = x$.
So, the old average was 30. The question asks for the "new average," so we calculate $30 + 5 = 35$.
This logic ensures that the total increase in runs (60) is distributed across all 12 innings, raising each one's average contribution by 5.
Step 4: Final Answer:
The new average of the batsman is 35.