A batsman scores 26 runs in a particular innings and thereby increases his batting average from 14 runs to 15 runs per innings. In the same match (that is, in this innings), how many total runs must he score so that his average increases further to 19 runs per innings?

Introduction / Context:
This question involves cricket batting averages and shows how a single innings can change a player's average. The batsman starts with one average, scores some runs, and his average changes. We are then asked how many runs he must score (in total in that innings) to reach a higher target average. The problem tests your understanding of averages, total runs and number of innings.

Given Data / Assumptions:

• Before this innings, the batsman's average was 14 runs per innings.
• Let the number of innings he had already played be n.
• So before this innings, total runs = 14n.
• During this innings, he scores 26 runs, after which his average becomes 15 runs per innings.
• This means after this innings he has played n + 1 innings.
• We must find how many runs he should score in this same innings in total so that his new average becomes 19 runs per innings.

Concept / Approach:
First, we use the information about the change from 14 to 15 to determine the number of innings n he had played before this match. Then we consider what total runs he must have after this innings to have an average of 19 over n + 1 innings. The total runs required minus the previous total gives the runs needed in this innings. The key ideas are expressing average as total runs divided by number of innings and solving simple linear equations.

Step-by-Step Solution:
Step 1: Let n be the number of innings before this innings. Total runs before = 14n. Step 2: After scoring 26 runs in this innings, total runs become 14n + 26 and the number of innings becomes n + 1. Step 3: Given that the average after scoring 26 runs is 15, we write (14n + 26) / (n + 1) = 15. Step 4: Cross multiply: 14n + 26 = 15n + 15. Step 5: Rearrange: 26 - 15 = 15n - 14n ⇒ 11 = n. Step 6: So before this innings he had played 11 innings and scored total runs 14 * 11 = 154. Step 7: To have an average of 19 runs per innings after this match, over n + 1 = 12 innings, total runs must be 19 * 12 = 228. Step 8: Let R be the total runs he must score in this innings. Then total runs after this innings are 154 + R, and we need 154 + R = 228. Step 9: Solve for R: R = 228 - 154 = 74. Step 10: Therefore, he must score 74 runs in this innings to raise his average to 19.

Verification / Alternative check:
If he scores 74 runs in this innings, his new total runs are 154 + 74 = 228 and number of innings is 12. His new average is 228 / 12 = 19 runs per innings, which matches the target. The earlier condition is also satisfied because after scoring just 26 runs, total would have been 180 and the average over 12 innings would be 15, consistent with the first given change in average.

Why Other Options Are Wrong:
If he scored 72 or 60 runs, his total runs would be lower than 228 and his average would remain below 19. If he scored 79 runs, total runs would be 233 and his average would be 233 / 12 ≈ 19.42, which is higher than 19. The given condition asks for the exact average of 19, not more. Therefore, only 74 runs satisfy the requirement exactly.

Common Pitfalls:
One common mistake is to forget to first determine the number of innings from the change in average from 14 to 15 and to work directly with the change from 15 to 19. Others might erroneously add 5 runs (the difference between 14 and 19) multiplied by some guessed number of innings. The reliable method is always to express each average situation in terms of total runs and number of innings and solve the resulting equations.

Final Answer:
The batsman must score a total of 74 runs in this innings to increase his average to 19 runs per innings.