Question

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.

Answer 1

It is given that the ratio between two positive numbers is `1/5:1/7`.

Let the two numbers be `1/5x` and `1/7x`.

The sum of the squares of the numbers = 666.

\[\Rightarrow \left(\frac{1}{5}x\right)^{2} + \left(\frac{1}{7}x\right)^{2} = 666\]

\[\Rightarrow \frac{1}{25}x^{2} + \frac{1}{49}x^{2} = 666\]

\[\Rightarrow \frac{1 \times 49}{25 \times 49}x^{2} + \frac{1 \times 25}{49 \times 25}x^{2} = 666\]

\[\Rightarrow \frac{49}{1,225}x^{2} + \frac{25}{1,225}x^{2} = 666\]

\[\Rightarrow \frac{74}{1,225}x^{2} = 666\]

\[\Rightarrow x^{2} = \frac{666 \times 1,225}{74}\]

\[\Rightarrow x^{2} = \frac{8,15,850}{74}\]

\[\Rightarrow x^{2} = 11,025\]

\[\Rightarrow x = \sqrt{11,025}\]

\[\Rightarrow x = 105\]

So, the numbers \[= \frac{1}{5}x = \frac{1}{5} \times 105 = 21\] and \[\frac{1}{7}x = \frac{1}{7} \times 105 = 15\]