CAT 2022 Slot 1 QA Q12 Number Systems

Question 12 of the Quantitative Ability (QA) section, CAT 2022 Slot 1, with CAT's official answer and a worked solution.

For any real number \(x\), let \([x]\) be the largest integer less than or equal to \(x\). If \(\sum_{n=1}^N\left[\frac{1}{5}+\frac{n}{25}\right]=25\) then \(N\) is

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

Answer: 44 Number Systems TITA

Solution

\(\left[\frac15 + \frac{n}{25}\right] = \left[\frac{n + 5}{25}\right]\).
This is 0 for \(n \le 19\), 1 for \(20 \le n \le 44\) and 2 from \(n = 45\).
Terms \(n = 20\) to \(44\) contribute 25 ones, so the sum reaches 25 exactly at \(N = 44\) (at \(N = 45\) it jumps to 27).