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

Introduction / Context:
This question tests your ability to manipulate linear equations in two variables and then convert that relationship into a ratio expression. Such algebraic manipulation appears frequently in aptitude exams, especially where variables are related by decimals, and you must express one in terms of the other and then substitute into a new expression.

Given Data / Assumptions:

• We are given the equation 1.5x = 0.04y.
• x and y are positive real numbers.
• We must find the value of (y - x) / (y + x).
• No specific numerical values of x and y are provided, so the expression must simplify to a constant.

Concept / Approach:
The key idea is to express y in terms of x, or x in terms of y, using the given linear equation. Once we know the ratio y / x, we can rewrite (y - x) / (y + x) in terms of that ratio alone. This removes the need to know individual values and turns the problem into pure algebraic simplification.

Step-by-Step Solution:
Start from the given equation: 1.5x = 0.04y.Rearrange to get y in terms of x: y = (1.5 / 0.04) * x.Compute the fraction 1.5 / 0.04 = 150 / 4 = 37.5, so y = 37.5x.Now form the required expression: (y - x) / (y + x) = (37.5x - x) / (37.5x + x).Simplify numerator and denominator: (36.5x) / (38.5x).Cancel x (since x is positive and nonzero): 36.5 / 38.5.Write these as fractions: 36.5 = 365 / 10 and 38.5 = 385 / 10.So the ratio becomes (365 / 10) / (385 / 10) = 365 / 385.Divide numerator and denominator by 5 to get 73 / 77.

Verification / Alternative check:
We can verify by choosing any convenient positive value for x and computing y from the relation. For example, let x = 2. Then y = 37.5 * 2 = 75. Now calculate (y - x) / (y + x) = (75 - 2) / (75 + 2) = 73 / 77. This matches the simplified algebraic result, confirming our answer is consistent and independent of the particular value of x chosen.

Why Other Options Are Wrong:
Option 71/77 is close but would come from a different ratio, and does not match the derived algebraic simplification. Option 36/37 would arise if the constants were different and is too large compared to the true ratio. Option 3/5 is a much simpler fraction and does not come from any correct manipulation of 1.5x = 0.04y. Option 37/73 is an inverted and slightly modified form and is not supported by any valid transformation of the given relation.

Common Pitfalls:
A common mistake is mishandling the decimal division 1.5 / 0.04, for example treating it as 1.5 / 4 or 0.15 / 0.04. Another error is substituting incorrectly into (y - x) / (y + x), such as forgetting to factor out x or cancel it. Some students also attempt to assign random values to x and y that do not satisfy the original equation, leading to inconsistent answers. Finally, forgetting to simplify the final fraction fully can cause you to miss the reduced form 73 / 77.

Final Answer:
The value of the expression (y - x) / (y + x), given that 1.5x = 0.04y, is 73/77.