If $$a : b = 2 : 3$$ and $$b : c = 2 : 3$$, then what is the value of $$(3a^2 + b^2 - c^2) : (a^2 + 2b^2 - c^2)$$?

$$a : b = 2 : 3$$ and $$b : c = 2 : 3$$

a : b : c = $$2\times2$$ : $$3\times2$$ :  $$3\times3$$

= 4 : 6 : 9

Let's assume a = 4y, b = 6y and c = 9y.

value of $$(3a^2 + b^2 - c^2) : (a^2 + 2b^2 - c^2)$$ = $$(3\left(4y\right)^2+\left(6y\right)^2-\left(9y\right)^2):(\left(4y\right)^2+2\left(6y\right)^2-\left(9y\right)^2)$$

= $$(3\times16y^2+36y^2 - 81y^2):(16y^2+2\times 36y^2- 81y^2)$$

= $$(48y^2+36y^2-81y^2):(16y^2+72y^2-81y^2)$$

= $$(84y^2-81y^2):(88y^2-81y^2)$$

= $$(3y^2):(7y^2)$$

= 3 : 7