Question

If $$x : y = 10 : 3$$, find $$(3x^{2} + 2y^{2}) : (3x^{2} - 2y^{2})$$.

Answer 1

We are given $$\frac{x}{y} = \frac{10}{3}$$, which gives $$\frac{x^2}{y^2} = \frac{100}{9}$$.

Divide the numerator and denominator of $$\frac{3x^2 + 2y^2}{3x^2 - 2y^2}$$ by $$y^2$$: $$\frac{3\left(\frac{x^2}{y^2}\right) + 2}{3\left(\frac{x^2}{y^2}\right) - 2} = \frac{3\left(\frac{100}{9}\right) + 2}{3\left(\frac{100}{9}\right) - 2}$$

Simplify: $$= \frac{\frac{100}{3} + 2}{\frac{100}{3} - 2}$$

$$= \frac{\frac{106}{3}}{\frac{94}{3}}$$

$$= \frac{106}{94}$$

$$= \frac{53}{47}$$

Therefore, $$(3x^2 + 2y^2) : (3x^2 - 2y^2) = 53 : 47$$.