CAT 2025 Slot 3 QA Question 7: 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 $0.7q$, the

Question

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 $0.7q$, the

Accepted Answer

Let $p+q=n$. Since $p+q+r = 900$, we have $n = 900-r$.
The conditions on $p$ are
$p \geq 0.3q\ 	ext{and}\ p \leq 0.7q$
$p \geq 0.3(n - p)$
$p \leq 0.7(n - p)$
$p \geq \dfrac{3n}{13}\ 	ext{and}\ p \leq \dfrac{7n}{17}$
The possible square values of $r$ are from $13^2 = 169$ to $22^2 = 484$.
For the

Let ppp, qqq and rrr be three natural numbers such that their sum is 900, and rrr is a perfect square whose value lies between 150 and 500. If ppp is not less than 0.3q0.3q0.3q and not more than 0.7q0.7q0.7q, then the sum of the maximum and minimum possible values of ppp is

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