If 15% of x is equal to 20% of y, what is the ratio x : y?

Introduction: This question tests your skill in forming and solving equations from percentage statements and then converting the relationship into a ratio. Such ratio problems based on percentages are very common in aptitude exams and help to check algebraic manipulation skills. Given Data / Assumptions: 15% of x is equal to 20% of y. We need to find the ratio x : y. Variables x and y are assumed to be positive real numbers. Concept / Approach: We translate the verbal statement into an equation using percentages in decimal form. Once we have an equation relating x and y, we can isolate the ratio x / y. Finally, we express this ratio in the simplest a : b form by simplifying the fraction. The percentages 15% and 20% become 0.15 and 0.20 respectively in the equation. Step-by-Step Solution: Given that 15% of x = 20% of y. Write 15% as 15 / 100 and 20% as 20 / 100. So the equation is (15 / 100) * x = (20 / 100) * y. Multiply both sides by 100 to remove denominators: 15 * x = 20 * y. We want x / y, so divide both sides of 15 * x = 20 * y by y and also by 15. This gives x / y = 20 / 15. Simplify 20 / 15 by dividing numerator and denominator by 5: 20 / 15 = 4 / 3. Therefore the ratio x : y is 4 : 3. Verification / Alternative check: We can verify with assumed numbers. Let y = 3 units. Then x must be 4 units according to the ratio 4 : 3. Compute 15% of x: 15% of 4 = 0.15 * 4 = 0.6. Compute 20% of y: 20% of 3 = 0.20 * 3 = 0.6. Since these are equal, the ratio 4 : 3 satisfies the original condition. Why Other Options Are Wrong: Ratios like 5 : 4, 4 : 5, 3 : 4, and 2 : 3 do not satisfy the equation 15% of x = 20% of y. For example, if x : y = 5 : 4, then 15% of 5 is 0.75 and 20% of 4 is 0.8, which are not equal. Only the ratio 4 : 3 balances the equation exactly. Common Pitfalls: Candidates sometimes reverse the ratio and report y : x instead of x : y, or they make a mistake when simplifying 20 / 15. Another common error is forgetting to convert percentages to fractions or decimals before forming the equation. Writing the relationship clearly and simplifying step by step avoids these problems. Final Answer: The required ratio is 4 : 3.