A cricketer has a certain batting average for 10 innings. In the 11th innings, he scores 108 runs, which increases his average by 6 runs. What is his new batting average after 11 innings?

Introduction / Context:
This is a classic cricket average problem. A batsman has a certain average for his first 10 innings. After scoring a big 108 runs in the 11th innings, his average increases by 6 runs. We are asked to find the new average after these 11 innings. Translating the verbal statement into algebraic equations is the key to solving it correctly.

Given Data / Assumptions:

• Let the original average after 10 innings be a runs per innings.
• Total runs in first 10 innings = 10a.
• Runs scored in 11th innings = 108.
• The average increases by 6 runs after the 11th innings, so new average = a + 6.
• Number of innings after the 11th = 11.
• We must find the new average a + 6.

Concept / Approach:
The fundamental relationship is: average = total runs / number of innings. We express the new total runs after 11 innings and then set that equal to 11 times the new average (because average times innings = total runs). Solving the resulting linear equation gives us the original average and then the new average.

Step-by-Step Solution:
Step 1: Let original average for 10 innings be a. Then total runs after 10 innings = 10a. Step 2: In the 11th innings, he scores 108 runs. New total runs after 11 innings = 10a + 108. Step 3: New average after 11 innings is a + 6. So, new average = (total runs after 11 innings) / 11. Therefore, (10a + 108) / 11 = a + 6. Step 4: Multiply both sides by 11 to clear the denominator. 10a + 108 = 11(a + 6). Step 5: Expand the right-hand side. 10a + 108 = 11a + 66. Step 6: Rearrange to solve for a. 108 − 66 = 11a − 10a → 42 = a. So original average a = 42 runs. Step 7: New average = a + 6 = 42 + 6 = 48 runs.
Verification / Alternative check:
Check total runs with a = 42: 10 innings give 10 * 42 = 420 runs. After 11th innings: total = 420 + 108 = 528 runs. Average after 11 innings = 528 / 11 = 48 runs, which is indeed 6 more than 42.
Why Other Options Are Wrong:
If the new average were 44, 46 or 49 runs, the corresponding total runs would not be consistent with a 108-run innings increasing the old average by exactly 6. For example, if the new average were 49, total runs after 11 innings would be 11 * 49 = 539, which is not equal to 420 + 108 = 528.
Common Pitfalls:
A frequent mistake is to assume that 108 is the new average or to try to add 6 directly to 108, which has no basis in the definition of average. Another error is to forget that the new average applies to all 11 innings, not just the last one.
Final Answer:
The cricketer's new batting average after 11 innings is 48 runs.