CAT 2022 Slot 2 QA Q8 Ratio, Mixtures & Averages

Question 8 of the Quantitative Ability (QA) section, CAT 2022 Slot 2, with CAT's official answer and a worked solution.

The average of a non-decreasing sequence of \(N\) numbers \(a_1, a_2, \ldots, a_N\) is 300 . If \(a_1\) is replaced by \(6 a_1\), the new average becomes 400 . Then, the number of possible values of \(a_1\) is

TITA: type in the answer. There are no options and no negative marking.

Answer: 14 Ratio, Mixtures & Averages TITA

Solution

The sum is \(300N\); replacing \(a_1\) by \(6a_1\) adds \(5a_1=100N\), so \(a_1=20N\).
\(a_1\) is the smallest term, so \(a_1\le300\), i.e. \(20N\le300\), \(N\le15\) (the rest average \(\frac{280N}{N-1}\ge20N\) also gives \(N\le15\)).
\(N=1\) fails (\(a_1\) would be both 300 and 20). So \(N=2,\dots,15\): 14 values of \(a_1\).