If $\frac{x}{2y} = \frac{6}{7}$, the value of $\frac{x - y}{x + y} + \frac{14}{19}$ is

### Concept & Formula This problem requires two steps: first, isolating the pure ratio of $\frac{x}{y}$ from a modified ratio, and second, substituting that pure ratio into a homogeneous algebraic fraction to evaluate a larger expression. $$ \text{Cross-multiplication: If } \frac{x}{ay} = \frac{b}{c}, \text{ then } \frac{x}{y} = \frac{a \cdot b}{c} $$ ### Step-by-Step Solution * **Given:** * $\frac{x}{2y} = \frac{6}{7}$ * Expression: $\frac{x - y}{x + y} + \frac{14}{19}$ * **Calculation:** 1. First, isolate the pure ratio $\frac{x}{y}$. Multiply both sides of the given equation by $2$. 2. $\frac{x}{y} = \frac{6 \cdot 2}{7} = \frac{12}{7}$. 3. Now, use the direct substitution method. Let $x = 12$ and $y = 7$. 4. Substitute these values into the fractional part of the target expression: $\frac{x - y}{x + y}$. 5. Numerator: $x - y = 12 - 7 = 5$. 6. Denominator: $x + y = 12 + 7 = 19$. 7. The fractional part evaluates to $\frac{5}{19}$. 8. Add this result to the remaining part of the original expression: $\frac{5}{19} + \frac{14}{19}$. 9. Since they share a common denominator, add the numerators: $\frac{5 + 14}{19} = \frac{19}{19}$. 10. Simplify to get $1$. ### Exam Strategy & Shortcut Always clean up the given ratio first. Trying to substitute $x = 6$ and $2y = 7$ (so $y = 3.5$) is a valid approach, but working with decimals like $3.5$ slows you down. Converting $\frac{x}{2y} = \frac{6}{7}$ into $\frac{x}{y} = \frac{12}{7}$ immediately gives you clean integers ($12$ and $7$) to plug into the rest of the formula. ### Common Pitfall A common trap is assuming $x = 6$ and $y = 7$ directly from $\frac{x}{2y} = \frac{6}{7}$, completely ignoring the $2$ in the denominator. This would yield $\frac{6-7}{6+7} = -\frac{1}{13}$, which would completely ruin the subsequent addition step. Always isolate $\frac{x}{y}$ first. ### Final Answer Therefore, the correct answer is 1.