Ratio equivalence: If 2A = 3B = 4C (all positive), determine the simplified ratio A : B : C.

Introduction / Context:
Problems that state an expression like 2A = 3B = 4C use a common equal value to relate different variables. The goal is to express A, B, and C as multiples of that common value and then reduce to the simplest whole-number ratio A : B : C.

Given Data / Assumptions:

• 2A = 3B = 4C (assume all quantities are positive for ratio interpretation).
• We want A : B : C in least integer terms.

Concept / Approach:
Let the common value be k. Then 2A = k, 3B = k, and 4C = k. Solve for A, B, C in terms of k, then scale to remove fractions and simplify.

Step-by-Step Solution:
Let 2A = 3B = 4C = k. Then A = k/2, B = k/3, C = k/4. A : B : C = (k/2) : (k/3) : (k/4). Cancel k and clear denominators by multiplying by LCM(2,3,4) = 12. Ratio becomes 12*(1/2) : 12*(1/3) : 12*(1/4) = 6 : 4 : 3.

Verification / Alternative check:
Pick k = 12 for convenience. Then A = 6, B = 4, C = 3. Check: 2A = 12, 3B = 12, 4C = 12; all equal. Thus A : B : C = 6 : 4 : 3 is consistent.

Why Other Options Are Wrong:

• 2 : 3 : 4 or permutations do not satisfy 2A = 3B = 4C simultaneously when scaled.
• 12 : 8 : 6 is correct in proportion but not simplified; the question expects the least integer ratio.

Common Pitfalls:

• Treating 2A, 3B, and 4C individually instead of introducing a common value k.
• Forgetting to clear denominators to get integer ratios.

Final Answer:
6 : 4 : 3