A batsman in his 14th inning makes a score of 91 runs, thereby increasing his…

For any collection of n data points, Total = Average × n. When one more data point is added and the average changes by a fixed amount, the new data point's value can be recovered from: New data point = (New average) × n − (Old average) × (n − 1). Equivalently, if the average simply increases by d after the n-th term is added, the average before that term equals (new term value) − n × d, and the resulting average after the term equals that value plus d.

1. Let A be the batsman's average over the first 13 innings (before the 14th).

2. Total runs scored in the first 13 innings = 13A.

3. After the 14th inning (score 91), total runs = 13A + 91.

4. Since the average after the 14th inning increases by 1, the new average = A + 1.

5. Total runs after 14 innings also equals 14 × (A + 1).

6. Equate the two totals: 13A + 91 = 14(A + 1).

7. Expand the right side: 13A + 91 = 14A + 14.

8. Collect terms: 91 − 14 = 14A − 13A, so A = 77.

9. The average after the 14th inning = A + 1 = 77 + 1 = 78.

Verify by totals: 14 innings at an average of 78 give a total of 14 × 78 = 1092 runs. Removing the 91 runs of the 14th inning leaves 1092 − 91 = 1001 runs from the first 13 innings, giving an average of 1001 ÷ 13 = 77 — exactly 1 run less than 78, confirming the increase condition stated in the question.

So the batsman's average after the 14th inning is 78 runs.