Find the compounded ratio of :

(i) (a - b) : (a + b) and (b² + ab) : (a² - ab)

(ii) (x + y) : (x - y), (x² + y²) : (x + y)² and (x² - y²)² : (x⁴ - y⁴)

(iii) (x² - 25) : (x² + 3x - 10); (x² - 4) : (x² + 3x + 2) and (x + 1) : (x² + 2x).

(i) Calculating the compounded ratio :

$\Rightarrow \dfrac{a - b}{a + b} \times \dfrac{b^2 + ab}{a^2 - ab} \\[1em] \Rightarrow \dfrac{a - b}{a + b} \times \dfrac{b(a + b)}{a(a - b)} \\[1em] \Rightarrow \dfrac{b}{a} \\[1em] \Rightarrow b : a.$

Hence, compounded ratio of (a - b) : (a + b) and (b² + ab) : (a² - ab) = b : a.

(ii) Calculating the compounded ratio :

$\Rightarrow \dfrac{x + y}{x - y} \times \dfrac{x^2 + y^2}{(x + y)^2} \times \dfrac{(x^2 - y^2)^2}{(x^4 - y^4)} \\[1em] \Rightarrow \dfrac{1}{x - y} \times \dfrac{x^2 + y^2}{(x + y)} \times \dfrac{(x^2 - y^2)^2}{(x^2 - y^2)(x^2 + y^2)} \\[1em] \Rightarrow \dfrac{1}{x - y} \times \dfrac{\cancel{x^2 + y^2}}{(x + y)} \times \dfrac{(x^2 - y^2)^{\cancel{2}}}{\cancel{(x^2 - y^2)}\cancel{(x^2 + y^2)}} \\[1em] \Rightarrow \dfrac{(x^2 - y^2)}{(x - y)(x + y)} \\[1em] \Rightarrow \dfrac{(x^2 - y^2)}{(x^2 - y^2)} \\[1em] \Rightarrow 1 : 1.$

Hence, the compounded ratio = 1 : 1.

(iii) Calculating the compounded ratio :

$\Rightarrow \dfrac{x^2 - 25}{x^2 + 3x - 10} \times \dfrac{x^2 - 4}{x^2 + 3x + 2} \times \dfrac{x + 1}{x^2 + 2x} \\[1em] \Rightarrow \dfrac{x^2 - 25}{x^2 + 5x - 2x - 10} \times \dfrac{(x - 2)(x + 2)}{x^2 + x + 2x + 2} \times \dfrac{x + 1}{x(x + 2)} \\[1em] \Rightarrow \dfrac{(x - 5)(x + 5)}{x(x + 5) - 2(x + 5)} \times \dfrac{(x - 2)(x + 2)}{x(x + 1) + 2(x + 1)} \times \dfrac{x + 1}{x(x + 2)} \\[1em] \Rightarrow \dfrac{(x - 5)(x + 5)}{(x - 2)(x + 5)} \times \dfrac{(x - 2)(x + 2)}{(x + 1)(x + 2)} \times \dfrac{x + 1}{x(x + 2)} \\[1em] \Rightarrow \dfrac{(x - 5)\cancel{(x + 5)}}{\cancel{(x - 2)}\cancel{(x + 5)}} \times \dfrac{\cancel{(x - 2)}\cancel{(x + 2)}}{\cancel{(x + 1)}\cancel{(x + 2)}} \times \dfrac{\cancel{x + 1}}{x(x + 2)} \\[1em] \Rightarrow \dfrac{(x - 5)}{x(x + 2)} \\[1em] \Rightarrow (x - 5) : x(x + 2).$

Hence, the compounded ratio = (x - 5) : x(x + 2).