After 10 innings the average score per innings of a batsman was 52. After 12 innings the average rose to 54. If the batsman had scored 16 more runs in the 12 th innings than in the previous one, how many runs did he score in the 11 th innings?
56
Calculating Batsman's Runs in 11th Inning
This problem involves calculating the runs scored by a batsman in a specific inning based on changes in their average score over several innings. We are given the average score after 10 innings and after 12 innings, along with a relationship between the runs scored in the 11th and 12th innings. We need to find the runs scored in the 11th inning.
Understanding Average Score
The average score is calculated by dividing the total runs scored by the number of innings played. So, Total Runs = Average Score × Number of Innings.
Step-by-Step Calculation
Step 1: Calculate the total runs after 10 innings.
The average score after 10 innings was 52.
Total runs after 10 innings = Average after 10 innings × Number of innings
Total runs after 10 innings = \(52 \times 10 = 520\) runs.
Let's denote the total runs after 10 innings as \(S_{10}\).
\(S_{10} = 520\)
Step 2: Calculate the total runs after 12 innings.
The average score after 12 innings rose to 54.
Total runs after 12 innings = Average after 12 innings × Number of innings
Total runs after 12 innings = \(54 \times 12\).
Let's calculate \(54 \times 12\):
| 50 | 4 | |
|---|---|---|
| 10 | 500 | 40 |
| 2 | 100 | 8 |
\(500 + 40 + 100 + 8 = 648\)
So, total runs after 12 innings = 648 runs.
Let's denote the total runs after 12 innings as \(S_{12}\).
\(S_{12} = 648\)
Step 3: Find the total runs scored in the 11th and 12th innings combined.
The difference between the total runs after 12 innings and the total runs after 10 innings is the sum of the runs scored in the 11th and 12th innings.
Runs in 11th inning + Runs in 12th inning = Total runs after 12 innings - Total runs after 10 innings
Let \(R_{11}\) be the runs scored in the 11th inning and \(R_{12}\) be the runs scored in the 12th inning.
\(R_{11} + R_{12} = S_{12} - S_{10}\)
\(R_{11} + R_{12} = 648 - 520\)
\(648 - 520 = 128\)
So, \(R_{11} + R_{12} = 128\).
Step 4: Use the given relationship between the 11th and 12th inning scores.
The batsman scored 16 more runs in the 12th inning than in the previous one (the 11th inning).
\(R_{12} = R_{11} + 16\)
Step 5: Solve the equations to find the runs in the 11th inning.
We have two equations:
- \(R_{11} + R_{12} = 128\)
- \(R_{12} = R_{11} + 16\)
Substitute the second equation into the first equation:
\(R_{11} + (R_{11} + 16) = 128\)
\(2 R_{11} + 16 = 128\)
Subtract 16 from both sides:
\(2 R_{11} = 128 - 16\)
\(2 R_{11} = 112\)
Divide by 2:
\(R_{11} = \frac{112}{2}\)
\(R_{11} = 56\)
So, the batsman scored 56 runs in the 11th inning.
To verify, let's find \(R_{12}\):
\(R_{12} = R_{11} + 16 = 56 + 16 = 72\)
Check if \(R_{11} + R_{12} = 128\):
\(56 + 72 = 128\). This is correct.
The runs scored in the 11th inning is 56.
Summary of Runs and Averages
| Innings | Total Runs | Average Score |
|---|---|---|
| After 10 | 520 | 52 |
| Runs in 11th | 56 | - |
| Total after 11 | \(520 + 56 = 576\) | \(576 / 11 \approx 52.36\) |
| Runs in 12th | 72 | - |
| Total after 12 | \(576 + 72 = 648\) | \(648 / 12 = 54\) |
The calculated values match the information given in the question, confirming our answer for the 11th inning score.
Revision Table: Key Concepts
| Concept | Definition/Formula | Application in Problem |
|---|---|---|
| Average Score | Total Runs / Number of Innings | Used to find total runs from average and innings played. |
| Total Runs | Average Score × Number of Innings | Calculated at different stages (after 10 and 12 innings). |
| Difference in Total Runs | Total Runs (later innings) - Total Runs (earlier innings) | Represents the sum of runs in the innings played between the two points. |
| Algebraic Equation | Representing unknown values (runs) and relationships with variables. | Used to set up and solve for the runs in the 11th inning. |
Additional Information: Batsman Averages
Calculating batsman averages is a fundamental concept in cricket statistics. It helps evaluate a player's performance over time. Here are some related points:
- An average represents the central tendency of a player's scores.
- A higher average generally indicates better batting performance.
- A player's average can change significantly based on their scores in recent innings. High scores will increase the average, while low scores will decrease it.
- In some statistical analyses, averages might be calculated excluding "not out" innings, but in basic problems like this, all innings where the batsman batted are typically included.
- Understanding how adding a new score affects the existing average is a common type of problem. If the new score is higher than the current average, the average will increase. If it's lower, the average will decrease. If it's equal, the average remains the same.
Important Questions from Average
The average of 28 numbers is 77. The average of first 14 numbers is 74 and the average of last 15 numbers is 84. If the 14 th number is excluded, then what is the average of remaining numbers? (correct to one decimal places)
24 students collected money for donation. The average contribution was Rs. 50. Later on, their teacher also contributed some money. Now the average contribution is Rs. 56. The teacher’s contribution is:
Out of 6 numbers, the sum of the first 5 numbers is 7 times the 6 th number. If their average is 136, then the 6 th number is:
The average of five numbers is 30. If one number is excluded, then average becomes 31. What is the excluded number?
The average weight of 49 students in a class is 39 kg. Seven of them whose average weight is 40 kg leave the class and other seven students whose average weight is 54 kg join the class. What is the new average weight (in kg) of the class?
