Ratio of sides of rectangle

Let the sides of the rectangle be $a$ and $b$. Area $= ab$. Perimeter $= 2(a + b) \implies$ Square of perimeter $= 4(a + b)^2$.

Given: $\frac{ab}{4(a+b)^2} = \frac{1}{25}$ $25ab = 4(a^2 + 2ab + b^2)$ $4a^2 - 17ab + 4b^2 = 0$ $(4a - b)(a - 4b) = 0$

For $a \le b$, $b = 4a \implies a : b = 1 : 4$.