If 1.5x = 0.04y for two real numbers x and y, then what is the value of the expression (y - x) / (y + x)?

Introduction / Context: This question tests algebraic manipulation and understanding of ratios between two variables. We are given a linear relationship between x and y and asked to evaluate the ratio (y - x) / (y + x) without knowing the actual values of x and y. Such questions appear frequently in aptitude tests to check symbolic reasoning skills.

Given Data / Assumptions:

• Relation between variables: 1.5x = 0.04y.
• We need to compute the value of (y - x) / (y + x).
• x and y are nonzero real numbers to keep the expressions meaningful.

Concept / Approach: From the equation 1.5x = 0.04y, we can find the ratio y / x. Once that ratio is known, we express y in terms of x and substitute into (y - x) / (y + x). The variable x cancels out, leaving a pure numerical fraction that can be simplified.

Step-by-Step Solution: Step 1: Start with 1.5x = 0.04y. Step 2: Rearrange to get y / x = 1.5 / 0.04. Step 3: Compute 1.5 / 0.04 = 150 / 4 = 37.5, so y = 37.5x. Step 4: Substitute into (y - x) / (y + x): (37.5x - x) / (37.5x + x). Step 5: Simplify numerator and denominator: (36.5x) / (38.5x). Step 6: Cancel x: (y - x) / (y + x) = 36.5 / 38.5. Step 7: Multiply top and bottom by 10: 36.5 / 38.5 = 365 / 385. Step 8: Divide both numerator and denominator by 5: 365 / 385 = 73 / 77.

Verification / Alternative check: Take a convenient value of x, for example x = 2. Then y = 37.5 × 2 = 75. Now compute (y - x) / (y + x) = (75 - 2) / (75 + 2) = 73 / 77. This matches the derived fraction, confirming the result.

Why Other Options Are Wrong: 7.3/77: This incorrectly places the decimal in the numerator and does not arise from correct simplification. 730/77 and 7300/77: These are much larger ratios and would require y to be enormously bigger relative to x, which is not implied by the equation. 7/77: This is a very small ratio and not consistent with the correct derivation.

Common Pitfalls: Mistakes often occur when dividing decimals, such as mishandling 1.5 / 0.04. Forgetting to cancel x in numerator and denominator can also lead to unnecessarily complex expressions. Carefully converting decimals to fractions helps avoid these errors.

Final Answer: The value of the expression (y - x) / (y + x) is 73/77.