Linked fractional equalities to overall ratio: If (1/2) of A = (2/5) of B = (1/3) of C, find the ratio A : B : C in simplest whole-number terms.

Introduction / Context:
When multiple proportional equalities are chained through constants on each variable, parametrization is the cleanest method. Here, three fractions of A, B, and C are equal, and we must convert that into a simple ratio A : B : C.

Given Data / Assumptions:
(1/2)A = (2/5)B = (1/3)C = k (say), with all variables positive.

Concept / Approach:
Express each variable in terms of k, then write the ratio and simplify to the least integers. Specifically: A = 2k, B = (5/2)k, C = 3k.

Step-by-Step Solution:
From (1/2)A = k ⇒ A = 2k. From (2/5)B = k ⇒ B = (5/2)k. From (1/3)C = k ⇒ C = 3k. Therefore, A : B : C = 2 : (5/2) : 3. Multiply by 2 to clear the fraction → 4 : 5 : 6.

Verification / Alternative check:
Let k = 2 for convenience. Then A = 4, B = 5, C = 6. Check: (1/2)A = 2; (2/5)B = 2; (1/3)C = 2; all equal, confirming the ratio.

Why Other Options Are Wrong:
Permutations like 6 : 4 : 5 or 5 : 4 : 6 scramble the order. The ratio must align with A, B, C derived from their respective equations.

Common Pitfalls:
Mixing up which constant multiplies which variable and failing to eliminate fractional components correctly.

Final Answer:
4 : 5 : 6