Smallest possible lowest score of batsman

CAT 2020 Slot 3 · QA · Medium · Averages

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A batsman played n+2n + 2 innings and got out on all occasions. His average score in these n+2n + 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 nn innings. In these nn innings, his average score was 30 runs and lowest score was xx runs. The smallest possible value of xx is

  1. A.

    1

  2. B.

    3

  3. C.

    2

  4. D.

    4

Answer

C

Explanation

Total runs in n+2n + 2 innings = 29(n+2)29(n + 2). Total runs in first nn innings = 30n30n. So 29(n+2)=30n+38+15  ⟹  29n+58=30n+53  ⟹  n=529(n+2) = 30n + 38 + 15 \implies 29n + 58 = 30n + 53 \implies n = 5.

Total runs in first 5 innings = 30×5=15030 \times 5 = 150. Maximum score in any of the first 5 innings is at most 37 (since < 38). To minimize xx (the lowest score), maximize the other 4 scores by making them equal to 37. x+4×37=150  ⟹  x+148=150  ⟹  x=2x + 4 \times 37 = 150 \implies x + 148 = 150 \implies x = 2.

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