CAT 2025 Slot 3 QA Q21 Inequalities & Modulus

Question 21 of the Quantitative Ability (QA) section, CAT 2025 Slot 3, with CAT's official answer and a worked solution.

Let p, q and r be three natural numbers such that their sum is 900, and r is a perfect square whose value lies between 150 and 500. If p is not less than 0.3q and not more than 0.7q, then the sum of the maximum and minimum possible values of p is

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

Answer: 397 Inequalities & Modulus TITA

Solution

\(r\) is one of \(13^2 = 169, \dots, 22^2 = 484\); let \(S = p + q = 900 - r\), so \(416 \le S \le 731\).
\(p \le 0.7q = 0.7(S - p) \Rightarrow p \le \frac{7S}{17}\); with \(S = 731 = 17 \times 43\), max \(p = 301\) (\(q = 430\)).
\(p \ge 0.3(S - p) \Rightarrow p \ge \frac{3S}{13}\); with \(S = 416 = 13 \times 32\), min \(p = 96\) (\(q = 320\)).
Sum \(= 301 + 96 = 397\).