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CAT 2022 Slot 3 QA Question & Solution

ArithmeticHard

Question

In an examination, the average marks of students in sections A and B are 32 and 60, respectively. The number of students in section A is 10 less than that in section B. If the average marks of all the students across both the sections combined is an integer, then the difference between the maximum and minimum possible number of students in section A is

Solution

Let the number of students in section A and B be a and b, respectively.

It is given, a = b - 10

$\ \ \frac{\ 32a+60b}{a+b}$ is an integer

$\ \ \frac{\ 32a+60\left(a+10\right)}{a+a+10}=k$

$\ \ \frac{\ 46a+300}{a+5}=k$

$k=\ \frac{\ 46\left(a+5\right)}{a+5}+\frac{70}{a+5}$

$k=\ \ 46+\frac{70}{a+5}$

a can take values 2, 5, 9, 30, 65

Difference = 65 - 2 = 63