The ratio between two positive numbers is $\dfrac{1}{5}:\dfrac{1}{7}$. If the sum of the squares of the numbers is 666, find the numbers.

It is given that the ratio between two positive numbers is $\dfrac{1}{5}:\dfrac{1}{7}$.

Let the two numbers be $\dfrac{1}{5}x$ and $\dfrac{1}{7}x$.

The sum of the squares of the numbers = 666.

$\Rightarrow \Big(\dfrac{1}{5}x\Big)^2 + \Big(\dfrac{1}{7}x\Big)^2 = 666\\[1em] \Rightarrow \dfrac{1}{25}x^2 + \dfrac{1}{49}x^2 = 666\\[1em] \Rightarrow \dfrac{1 \times 49}{25 \times 49}x^2 + \dfrac{1 \times 25}{49 \times 25}x^2 = 666\\[1em] \Rightarrow \dfrac{49}{1,225}x^2 + \dfrac{25}{1,225}x^2 = 666\\[1em] \Rightarrow \dfrac{74}{1,225}x^2 = 666\\[1em] \Rightarrow x^2 = \dfrac{666 \times 1,225}{74}\\[1em] \Rightarrow x^2 = \dfrac{8,15,850}{74}\\[1em] \Rightarrow x^2 = 11,025\\[1em] \Rightarrow x = \sqrt{11,025}\\[1em] \Rightarrow x = 105$

So, the numbers = $\dfrac{1}{5}x = \dfrac{1}{5} \times 105$ = 21 and $\dfrac{1}{7}x = \dfrac{1}{7} \times 105$ = 15

Hence, the two numbers = 21 and 15.