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.
Similar Questions
The mean of 5 numbers is 15. If one more number is included, the mean of 6 numbers becomes 17. What is the included number?
In an asymmetrical distribution, if the mean and median of the distribution are 270 and 220 respectively, then the mode of the data is
Average marks in Mathematics of Section A comprising 30 students is 65 and that of Section B comprising 35 students is 70. What are the average marks (approximately) of both the sections if it was detected later that an entry of 47 marks was wrongly made as 74 ?
The geometric mean of x and y is 6 and the geometric mean of x, y and z is also 6. Then the value of z is
The average height of 22 students of a class is 140 cm and the average height of 28 students of another class is 152 cm. What is the average height of students of both classes?
If the average of 9 consecutive positive integers is 55, then what is the largest integer?
In a class of 100 students, there are 70 boys whose average marks in a subject are 75. If average marks of the complete class is 72, then what is the average marks of the girls?
The average of 50 consecutive natural numbers is x. What will be the new average when the next four numbers are also included?
Consider two-digit numbers which remain the same when the digits interchange their positions. What is the average of such two-digit numbers?
The arithmetic mean of two numbers is 10 and their geometric mean is 8. What are the two numbers?
Important Questions from Average
Average of 40 numbers is 71, if the number 100 replaced by 140, then average is increased by
The captain of a football team of 11 members is 28 years old and the goalkeeper is 4 years older than him. If the ages of these two are removed, then the average age of the remaining players is two years less than the average age of the whole team. What is the average age of the team?
There are two Classes A and B having 25 and 30 students respectively. In Class-A the highest score is 21 and lowest score is 17. In Class-B the highest score is 30 and lowest score is 22. Four students are shifted from Class-A to Class-B.
Consider the following statements:
1. The average score of Class-B will definitely decrease.
2. The average score of Class-A will definitely increase.
Which of the above statements is/are correct?
The average weight of A, B, Cis 40 kg, the average weight of B, D, Eis 42 kg and the weight of Fis equal to that of B. What is the average weight of A, B, C, D, Eand F?
A cow costs more than 4 goats but less than 5 goats. If a goat costs between Rs. 600 and Rs. 800, which of the following is a most valid conclusion?
