Question

If $$a : b = 2 : 5$$, find $$(3a^2 - 2ab + 5b^2) : (a^2 + 7ab - 2b^2)$$.

Answer 1

Let $$a = 2k$$ and $$b = 5k$$.

Calculate the numerator $$3a^2 - 2ab + 5b^2$$:

$$3(2k)^2 - 2(2k)(5k) + 5(5k)^2$$

$$= 3(4k^2) - 20k^2 + 5(25k^2)$$

$$= 12k^2 - 20k^2 + 125k^2$$

$$= 117k^2$$

Calculate the denominator $$a^2 + 7ab - 2b^2$$:

$$(2k)^2 + 7(2k)(5k) - 2(5k)^2$$

$$= 4k^2 + 70k^2 - 50k^2$$

$$= 24k^2$$

Find the ratio: $$\frac{117k^2}{24k^2} = \frac{117}{24} = \frac{39}{8}$$

Therefore, the ratio is $$39 : 8$$.