Two natural numbers are in the ratio 3 : 4 . Find the numbers if the difference between their squares is 175.

Since, the numbers are in the ratio 3 : 4, hence the numbers be 3x and 4x.

Given , difference between their squares = 175

∴ (4x)² - (3x)² = 175

$\Rightarrow 16x^2 - 9x^2 = 175 \\[1em] \Rightarrow 7x^2 = 175 \\[1em] \Rightarrow x^2 = \dfrac{175}{7} \\[1em] \Rightarrow x^2 = 25 \\[1em] \Rightarrow x^2 - 25 = 0 \\[1em] \Rightarrow (x - 5)(x + 5) = 0 \\[1em] \Rightarrow x - 5 = 0 \text{ or } x + 5 = 0 \\[1em] x = 5 \text{ or } x = -5$

Since, numbers are natural hence, x ≠ -5.

∴ x = 5, 3x = 15 , 4x = 20.

Hence, the required numbers are 15, 20.