Question

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 lowest score was x runs. The smallest possible value of x is

• A. 2
• B. 3
• C. 4
• D. 1

Answer 1

The Correct Option is A

To determine the minimum possible value for \(x\), representing the lowest score across the initial \(n\) innings, adhere to the following procedure:

1. Calculate total runs across \(n+2\) innings: With an average score of 29, the aggregate runs are \((n+2) \times 29 = 29n + 58\).
2. Determine total runs in the final 2 innings: The scores were 38 and 15, totaling 38 + 15 = 53.
3. Find runs scored in the first \(n\) innings: Subtract the last 2 innings' runs from the overall total: \(29n + 58 - 53 = 29n + 5\).
4. Establish the average score for the first \(n\) innings: This is given as 30, implying a total of \(30n\) runs.
5. Equate the run totals for the first \(n\) innings: Setting \(29n + 5 = 30n\) yields \(30n - 29n = 5\), thus \(n = 5\).
6. Let the scores in the first \(n\) innings be \(a_1, a_2, ..., a_n\). With \(n = 5\) and an average of 30, the sum is \(a_1 + a_2 + ... + a_5 = 150\).
7. Given that all scores are below 38 and \(\min(a_1, a_2, ..., a_5) = x\), to minimize \(x\), assign scores as close to 38 as possible. Consider the set: 37, 37, 37, 37, and \(x\). The sum becomes: \(4 \times 37 + x = 150\).
8. Solve for \(x\): \(148 + x = 150\) results in \(x = 2\).

Consequently, the smallest achievable value for \(x\) is 2.