If $\frac{x}{y} = \frac{6}{5}$, then the value of $\left(\frac{6}{7} - \frac{5x - y}{5x + y}\right)$ is equal to

### Concept & Formula This problem tests the concept of ratio substitution in homogeneous algebraic expressions. Since the algebraic fraction $\frac{5x - y}{5x + y}$ has the same degree for variables in both the numerator and denominator, we can directly substitute the proportional values of $x$ and $y$. $$ \text{If } \frac{x}{y} = \frac{a}{b}, \text{ assume } x = a \text{ and } y = b \text{ for homogeneous fractions.} $$ ### Step-by-Step Solution * **Given:** * $\frac{x}{y} = \frac{6}{5}$ * Expression: $\frac{6}{7} - \frac{5x - y}{5x + y}$ * **Calculation:** 1. Based on the given ratio, directly assume proportional values: let $x = 6$ and $y = 5$. 2. Substitute these values into the algebraic fraction part of the expression: $\frac{5x - y}{5x + y}$. 3. Numerator: $5(6) - 5 = 30 - 5 = 25$. 4. Denominator: $5(6) + 5 = 30 + 5 = 35$. 5. The fraction becomes $\frac{25}{35}$. 6. Simplify this fraction by dividing the top and bottom by $5$: $\frac{5}{7}$. 7. Substitute this simplified value back into the full expression: $\frac{6}{7} - \frac{5}{7}$. 8. Since the denominators are the same, simply subtract the numerators: $\frac{6 - 5}{7} = \frac{1}{7}$. ### Exam Strategy & Shortcut For any equation where you are given a ratio like $x/y = 6/5$ and asked to solve a fraction where every term contains either $x$ or $y$ to the power of 1, skip finding a constant $k$. Immediately plug in $x=6$ and $y=5$. This eliminates algebraic manipulation and turns it into basic arithmetic. ### Common Pitfall A common error is trying to divide the numerator and denominator by $y$ to force the expression into forms of $(x/y)$. While mathematically correct, it often leads to messy fraction-within-fraction calculations like $\frac{5(6/5) - 1}{5(6/5) + 1}$, which increases the likelihood of a calculation error under time pressure. Direct substitution is safer. ### Final Answer Therefore, the correct answer is $\frac{1}{7}$.