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CAT 2022 Slot 2 — QA Question 10

Change in AverageEasy

The average of a non-decreasing sequence of N number a1, a2, ..., aN is 300. If a1 is replaced by 6a1, the new average becomes 400. Then, the number of possible values of a1 is

Answer & solution

Answer: 14

Solution

Easy

Subtract the two "sum = average ×\times count" equations to isolate a1a_1 in terms of NN. Then use the fact that the list is non-decreasing (so a1a_1 is the smallest term, hence a1≤a_1\le average) plus a quick sanity check at the small end to count the valid NN.

1

Two total-sum equations (average ×N\times N):

a1+a2+⋯+aN=300N(1)6a1+a2+⋯+aN=400N(2)\begin{aligned} &a_1+a_2+\dots+a_N=300N \quad(1)\\ &6a_1+a_2+\dots+a_N=400N \quad(2) \end{aligned}
2

Subtract (1)(1) from (2)(2):

5a1=100N  ⇒  a1=20N.5a_1=100N \;\Rightarrow\; a_1=20N.
3

Bound NN. Since the sequence is non-decreasing, a1a_1 is the smallest term, so it cannot exceed the average 300300:

a1=20N≤300  ⇒  N≤15.a_1=20N\le300 \;\Rightarrow\; N\le15.
4

Drop N=1N=1. If N=1N=1 there is a single number and its average is itself, which cannot be both 300300 and 400400. So NN ranges over 2,3,…,152,3,\dots,15:

15−2+1=14 values.15-2+1=14\ \text{values.}

Number of possible values of a1a_1 =14=\mathbf{14}.

CAT 2022 Slot 2 QA Q10: The average of a non-decreasing sequence of N number a 1 , a 2 , ..., a N is 300. If a1 is replaced by 6a 1 , — Solution | TheCATExam