If $\frac{a}{b} = \frac{4}{5}$ and $\frac{b}{c} = \frac{15}{16}$, then $\frac{c^2 - a^2}{c^2 + a^2}$ would be

### Concept & Ratio Combination To evaluate an expression involving multiple variables linked by chained ratios, first combine the ratios into a single relationship between the required variables. For $a$, $b$, and $c$, finding the direct ratio of $a : c$ allows for immediate substitution since the given algebraic expression is homogeneous. ### Step-by-Step Solution - **Given:** $\frac{a}{b} = \frac{4}{5}$ and $\frac{b}{c} = \frac{15}{16}$. - **Find direct ratio a/c:** Multiply the two given ratios to eliminate $b$. $\frac{a}{c} = \frac{a}{b} \times \frac{b}{c} = \frac{4}{5} \times \frac{15}{16}$ - **Simplify:** $\frac{a}{c} = \frac{4 \times 15}{5 \times 16} = \frac{1 \times 3}{1 \times 4} = \frac{3}{4}$. - **Substitute:** Since the expression $\frac{c^2 - a^2}{c^2 + a^2}$ is homogeneous (all terms are degree 2), let $a = 3$ and $c = 4$. - **Calculate:** - Numerator: $c^2 - a^2 = 4^2 - 3^2 = 16 - 9 = 7$ - Denominator: $c^2 + a^2 = 4^2 + 3^2 = 16 + 9 = 25$ - **Result:** The ratio is $\frac{7}{25}$. ### Exam Strategy & Shortcut When you see chained fractions like $a/b$ and $b/c$, and you need a relation between $a$ and $c$, immediately multiply them. Recognize the Pythagorean triplet numbers $(3, 4, 5)$ which makes squaring $(16 - 9 = 7$ and $16 + 9 = 25)$ instant mental math. ### Common Pitfall A common mistake is attempting to make $b$ the same in both ratios (e.g., changing $4/5$ to $12/15$) and then using $a=12$, $b=15$, $c=16$. While mathematically correct, squaring 12 and 16 takes much longer than using the reduced ratio $a=3, c=4$. ### Final Answer Therefore, the correct answer is **$\frac{7}{25}$**.