Average, Mixture and Alligation Question 2 - CAT 2020
A batsman played n+2 innings and got out on all occasions. His average score in these n+2 innings was 29 runs and he scored 38 and 15 runs in the last two innings. The batsman scored less than 38 runs in each of the first n innings. In these n innings, his average score was 30 runs and the lowest score was x runs. The smallest possible value of x is:
Step-by-Step Explanation
- Analyze the Inning Totals:
The total runs scored across all n + 2 innings can be written by balancing the two given averages:
Total Runs = 30n + 38 + 15 = 29(n + 2)
30n + 53 = 29n + 58
n = 5 innings
- Optimize the Scores within the First n Innings:
The total runs scored in the first 5 innings = 30 × 5 = 150 runs.
To find the smallest possible value of the lowest score x, we must maximize the performance in the other 4 innings.
The problem states that the batsman scored strictly less than 38 runs in each of these first n innings. Since scores must be integers, the absolute maximum runs he could score in any individual inning is 37.
x + 37 × 4 = 150
x + 148 = 150 ⟹ x = 2 runs
Correct Option: C
💡 FundaMakers Tip:
Maximum/Minimum boundary questions depend entirely on absolute constraints. When given that values are strictly less than a benchmark (<38), immediately set the non-target variables to the next valid integer floor or ceiling (37). Maximizing the boundary elements forces the target variable x to drop cleanly to its absolute lower limit.