If $a : b = 5 : 7$, then find the ratio $\frac{6a^2 - 2b^2}{b^2 - a^2}$.

Question

Accepted Answer

**Solution:**

1. Given the ratio:
   $$a : b = 5 : 7$$
   Let $a = 5k$ and $b = 7k$ for some non-zero $k$.

2. Calculate the squares:
   $$a^2 = (5k)^2 = 25k^2$$
   $$b^2 = (7k)^2 = 49k^2$$

3. Substitute into the expression:
   $$\frac{6a^2 - 2b^2}{b^2 - a^2} = \frac{6(25k^2) - 2(49k^2)}{49k^2 - 25k^2} = \frac{150k^2 - 98k^2}{24k^2}$$

4. Simplify numerator and denominator:
   $$= \frac{52k^2}{24k^2} = \frac{52}{24}$$

5. Simplify the fraction:
   $$\frac{52}{24} = \frac{13}{6}$$

6. The ratio is therefore:
   $$13 : 6$$

**Answer:** $13 : 6$

Answer

Answer

Answer