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 \(54 \times 12\):

\(500 + 40 + 100 + 8 = 648\)

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:

1. \(R_{11} + R_{12} = 128\)
2. \(R_{12} = R_{11} + 16\)

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

The calculated values match the information given in the question, confirming our answer for the 11th inning score.

Revision Table: Key Concepts

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.