Brian Lara scored 6000 runs in a certain number of innings. In the next five…

Concept:

Average = Total runs ÷ Number of innings. In a change-in-average problem, express the average before and after the additional innings as two ratios in one unknown (the innings count), then equate them using the stated change in average.

Application:

1. Let x = the total number of innings Lara played in all.

2. Before the last five innings, he had played (x − 5) innings for 6000 runs, so his average then was 6000/(x − 5).

3. After the five innings in which he scored 90 runs in total, his total is 6000 + 90 = 6090 runs in x innings, so his average now is 6090/x.

4. Since the average fell by 2, set up: 6090/x = 6000/(x − 5) − 2.

5. Multiply both sides by x(x − 5): 6090(x − 5) = 6000x − 2x(x − 5).

6. Expand: 6090x − 30450 = 6000x − 2x2 + 10x.

7. Simplify and rearrange: 2x2 + 80x − 30450 = 0, i.e. x2 + 40x − 15225 = 0.

8. Solve with the quadratic formula: x = [−40 ± √(1600 + 60900)] / 2 = [−40 ± 250] / 2.

9. Discard the negative root (innings can't be negative): x = 210/2 = 105.

Cross-check:

With x = 105: before the last five innings he had 100 innings at 6000 runs, average 60; after, 105 innings at 6090 runs, average 58 — a fall of exactly 2, confirming the total is 105 innings.