If $x : y = 7 : 3$, then the value of $\frac{xy + y^2}{x^2 - y^2}$ is

### Concept & Algebraic Simplification Before substituting values, look for opportunities to factor and simplify the algebraic expression. This often reduces the size of the numbers you have to calculate. ### Step-by-Step Solution - **Given:** $x : y = 7 : 3$. Let $x = 7$ and $y = 3$. - **Expression:** $\frac{xy + y^2}{x^2 - y^2}$ - **Method 1: Direct Substitution** - Numerator: $7(3) + 3^2 = 21 + 9 = 30$ - Denominator: $7^2 - 3^2 = 49 - 9 = 40$ - Ratio: $\frac{30}{40} = \frac{3}{4}$ - **Method 2: Algebraic Simplification** - Factor numerator: $y(x + y)$ - Factor denominator: $(x - y)(x + y)$ - Simplify expression: $\frac{y(x + y)}{(x - y)(x + y)} = \frac{y}{x - y}$ - Substitute values: $\frac{3}{7 - 3} = \frac{3}{4}$ ### Exam Strategy & Shortcut Always check if an algebraic fraction can be simplified first. Factoring the expression to $\frac{y}{x-y}$ makes the calculation instantaneous: $\frac{3}{7-3} = \frac{3}{4}$, completely bypassing squares and larger numbers. ### Common Pitfall Using direct substitution without noticing algebraic identities like the difference of squares can consume more time and increase the risk of calculation errors on more complex questions. ### Final Answer Therefore, the correct answer is **$\frac{3}{4}$**.