Difficulty: Medium
Correct Answer: 15 : 10 : 6
Explanation:
Introduction / Context: Ratios involving reciprocals invert the usual proportionality. If 1/x : 1/y : 1/z is known, then x : y : z is proportional to the reciprocals of those terms. Converting correctly is vital for many algebraic and rate problems.
Given Data / Assumptions: 1/x : 1/y : 1/z = 2 : 3 : 5 with positive variables.
Concept / Approach: Let 1/x = 2k, 1/y = 3k, 1/z = 5k. Then x = 1/(2k), y = 1/(3k), z = 1/(5k). Therefore, x : y : z = (1/2) : (1/3) : (1/5). Clear denominators to get integers.
Step-by-Step Solution: x : y : z = (1/2) : (1/3) : (1/5). Multiply all by LCM(2,3,5) = 30. x : y : z = 30*(1/2) : 30*(1/3) : 30*(1/5) = 15 : 10 : 6.
Verification / Alternative check: Take x = 15, y = 10, z = 6. Then 1/x : 1/y : 1/z = (1/15) : (1/10) : (1/6) = 2 : 3 : 5 after multiplying by 30, confirming consistency.
Why Other Options Are Wrong: Options that permute the order (e.g., 6 : 15 : 10) or mix terms do not correspond to the correct inversion step from reciprocals.
Common Pitfalls: Forgetting that the ratio of reciprocals inverts the direction of proportionality. Always invert and then clear denominators carefully.
Final Answer: 15 : 10 : 6
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