A cricketer has certain average of 10 innings. In the 11 th inning, he scored 108 runs, thereby increasing his average by 6 runs. What is his new average?
48
Solving the Cricketer Average Problem
This question asks us to find the new average of a cricketer after scoring 108 runs in the 11th inning, which increased his previous average (after 10 innings) by 6 runs. We can solve this by setting up an equation based on the definition of the average.
Understanding Cricket Average
A cricketer's batting average is calculated by dividing the total number of runs scored by the total number of innings batted (where they were not out). In this problem, we are dealing with the total runs over a certain number of completed innings.
The formula for average is:
\[ \text{Average} = \frac{\text{Total Runs}}{\text{Number of Innings}} \]
Step-by-Step Solution for Cricketer's New Average
Let's define the variables:
- Let \(A_{10}\) be the cricketer's average after 10 innings.
- Let \(T_{10}\) be the total runs scored by the cricketer in the first 10 innings.
From the definition of average, we know:
\[ A_{10} = \frac{T_{10}}{10} \]
So, the total runs in the first 10 innings can be expressed as:
\[ T_{10} = 10 \times A_{10} \]
In the 11th inning, the cricketer scored 108 runs. So, the total runs after 11 innings (\(T_{11}\)) will be:
\[ T_{11} = T_{10} + 108 \]
Substitute the expression for \(T_{10}\):
\[ T_{11} = 10 \times A_{10} + 108 \]
The problem states that the average increased by 6 runs in the 11th inning. This means the new average after 11 innings (\(A_{11}\)) is:
\[ A_{11} = A_{10} + 6 \]
We can also calculate the new average using the total runs after 11 innings and the number of innings (which is 11):
\[ A_{11} = \frac{T_{11}}{11} \]
Now, we can set up an equation by substituting the expressions for \(A_{11}\) and \(T_{11}\):
\[ A_{10} + 6 = \frac{10 \times A_{10} + 108}{11} \]
To solve for \(A_{10}\), multiply both sides of the equation by 11:
\[ 11 \times (A_{10} + 6) = 10 \times A_{10} + 108 \]
Distribute the 11 on the left side:
\[ 11 \times A_{10} + 66 = 10 \times A_{10} + 108 \]
Subtract \(10 \times A_{10}\) from both sides to isolate the term with \(A_{10}\):
\[ 11 \times A_{10} - 10 \times A_{10} + 66 = 108 \]
\[ A_{10} + 66 = 108 \]
Subtract 66 from both sides to find the value of \(A_{10}\):
\[ A_{10} = 108 - 66 \]
\[ A_{10} = 42 \]
So, the cricketer's average after 10 innings was 42 runs.
The question asks for the new average, which is \(A_{11}\). We know that \(A_{11} = A_{10} + 6\).
\[ A_{11} = 42 + 6 \]
\[ A_{11} = 48 \]
The cricketer's new average after 11 innings is 48 runs.
Summary Table
| Item | Value/Expression |
|---|---|
| Number of Innings (Initial) | 10 |
| Initial Average (\(A_{10}\)) | \(A_{10}\) |
| Total Runs (Initial) | \(10 \times A_{10}\) |
| Runs in 11th Inning | 108 |
| Number of Innings (New) | 11 |
| Total Runs (New) | \(10 \times A_{10} + 108\) |
| New Average (\(A_{11}\)) | \(A_{10} + 6\) |
| New Average (Calculated) | \(\frac{10 \times A_{10} + 108}{11}\) |
| Equation | \(A_{10} + 6 = \frac{10 \times A_{10} + 108}{11}\) |
| Calculated Initial Average (\(A_{10}\)) | 42 |
| Calculated New Average (\(A_{11}\)) | 48 |
The calculated new average is 48.
Revision Table: Key Concepts in Average Calculation
| Concept | Definition | Formula | Application in Problem |
|---|---|---|---|
| Average | A measure of central tendency; the sum of values divided by the number of values. | \( \text{Average} = \frac{\text{Sum}}{\text{Count}} \) | Used to relate total runs and number of innings. |
| Total Sum (Runs) | The cumulative score over a number of innings. | \( \text{Total Sum} = \text{Average} \times \text{Count} \) | Used to find the total runs before and after the 11th inning. |
| Change in Average | How the average changes when new data (innings score) is added. | New Avg = Old Avg + Change | The 11th inning score caused a specific increase (+6) in the average. |
Additional Information on Cricket Statistics
Cricket statistics, like batting average, are crucial for evaluating player performance over time. While the basic average calculation is simple, more complex statistics exist, such as strike rate (runs per 100 balls faced) and economy rate (runs conceded per over for bowlers). These metrics provide a fuller picture of a player's impact on the game.
Understanding how adding new data points affects the average is a common type of quantitative problem. It often involves algebra to solve for an unknown initial value based on how the final average changed.
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