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 $0.7q$, then the sum of the maximum and minimum possible values of $p$ is

Hint:

When you have constraints like $ap \le q \le bp$ along with $p + q + r = \text{constant}$, try expressing $q$ in terms of $p$ and $r$, then convert the inequalities into bounds for $p$ in terms of $r$. After that, use monotonicity (increasing/decreasing behavior) to decide which extreme values of $r$ give the extreme values of $p$.

Answer 1

Correct Answer: 397

Step 1: Perfect squares strictly between 150 and 500 range from \( 13^2=169 \) to \( 22^2=484 \).
Step 2: With \( S=p+q=900-r \), the constraint gives \( \frac{3S}{13}\le p\le\frac{7S}{17} \). To maximise \( p \), take smallest \( r=169 \) (so \( S=731 \)): \( p_{max}=\frac{7(731)}{17}=301 \).
Step 3: To minimise \( p \), take largest \( r=484 \) (so \( S=416 \)): \( p_{min}=\frac{3(416)}{13}=96 \).
\[ \boxed{301+96=397} \]