If $1.5x = 0.04y$, then the value of $\left(\frac{y - x}{y + x}\right)$ is

### Concept & Strategy Find the ratio of $x$ to $y$ from the initial linear equation, then substitute those fractional values directly into the target rational expression. ### Step-by-Step Solution * Rearrange the given equation to isolate the ratio $\frac{x}{y}$: $$ 1.5x = 0.04y \implies \frac{x}{y} = \frac{0.04}{1.5} $$ * Eliminate the decimals by multiplying the numerator and denominator by $100$: $$ \frac{x}{y} = \frac{4}{150} = \frac{2}{75} $$ * We can assign proportional values: $x = 2$ and $y = 75$. * Substitute these values into the required target expression: $$ \frac{y - x}{y + x} = \frac{75 - 2}{75 + 2} $$ $$ = \frac{73}{77} $$ ### Exam Strategy & Shortcut Use componendo and dividendo logic rules. Once you establish $\frac{y}{x} = \frac{75}{2}$, the expression $\frac{y-x}{y+x}$ is simply $\frac{75-2}{75+2} = \frac{73}{77}$. This takes less than 15 seconds. ### Common Pitfall Accidentally reversing the ratio values as $\frac{x}{y} = \frac{75}{2}$, which leads to a sign error or choosing an incorrect distractor value. ### Final Answer **Therefore, the correct answer is $\frac{73}{77}$.**