A player Score Certain run/upto 9th-inning in 10 and 11 his total  is 108 by this his avg decrease, by: 12 run. find his total score upto 11th Innings

Let's solve the problem step by step:

1. **Initial Setup:**
   - Let the total score up to the 9th inning be \( S_9 \).
   - The player scores 10 runs in the 10th inning and 11 runs in the 11th inning, making his total score after 11 innings:
     \[
     S_{11} = S_9 + 10 + 11 = S_9 + 21
     \]

2. **Average Before and After the 10th and 11th Innings:**
   - Let the average score after the 9th inning be \( A \). Therefore:
     \[
     A = \frac{S_9}{9}
     \]
   - The average score after 11 innings is given to be \( A - 12 \). Therefore:
     \[
     A - 12 = \frac{S_{11}}{11} = \frac{S_9 + 21}{11}
     \]

3. **Set Up the Equation:**
   - We know:
     \[
     \frac{S_9}{9} = A \quad \text{and} \quad A - 12 = \frac{S_9 + 21}{11}
     \]
   - Substitute \( A = \frac{S_9}{9} \) into the second equation:
     \[
     \frac{S_9}{9} - 12 = \frac{S_9 + 21}{11}
     \]
   - Multiply both sides by 11 to eliminate the fraction on the right-hand side:
     \[
     \frac{11S_9}{9} - 132 = S_9 + 21
     \]
   - Multiply through by 9 to clear the fraction:
     \[
     11S_9 - 1188 = 9S_9 + 189
     \]
   - Simplify and solve for \( S_9 \):
     \[
     2S_9 = 1377
     \]
     \[
     S_9 = 688.5 \text{ runs}
     \]

4. **Calculate the Total Score up to 11th Inning:**
   - Now, add the scores of the 10th and 11th innings to find the total score up to the 11th inning:
     \[
     S_{11} = 688.5 + 21 = 709.5 \text{ runs}
     \]

So, the total score up to the 11th inning is **709.5 runs**.

Would you like further details or have any questions? Here are some related questions you might consider:

1. What would be the new average if the player scored 15 and 16 runs in the 10th and 11th innings instead?
2. What if the player's average had decreased by 15 runs instead of 12?
3. How would the problem change if the scores in the 10th and 11th innings were different?
4. What happens to the average if the player scores exactly his average in the 10th and 11th innings?
5. How does the total score up to 11th innings change if the scores in the 9th inning were doubled?

**Tip:** When dealing with averages, changes in the total number of innings and the scores in each inning can significantly impact the overall average.

Learn how to calculate the total score up to the 11th inning for a player with a decreased average after scoring certain runs in the 10th and 11th innings. This problem covers arithmetic and algebraic concepts related to averages. Suitable for Grades 6-8 students.

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