Ratio equivalence expanded: If 2A = 3B = 4C (all three expressions equal the same positive quantity), determine the simplified ratio A : B : C. Show clearly how you scale each term to obtain integers.

Introduction / Context:
Problems that present chained equalities such as 2A = 3B = 4C ask you to express each variable in terms of a single common value and then reduce to a clean integer ratio. This checks fluency with proportional scaling and the use of least common multiples to clear denominators neatly.

Given Data / Assumptions:

• 2A = 3B = 4C = k, where k is some positive common value.
• We seek A : B : C in simplest integer terms.

Concept / Approach:
Express each variable in terms of k, then remove fractions by multiplying all terms with the least common multiple of denominators. Finally, simplify to a primitive ratio (no common factor).

Step-by-Step Solution:

Verification / Alternative check:
Pick k = 12. Then A = 6, B = 4, C = 3. Indeed, 2A = 12, 3B = 12, 4C = 12—consistent.

Why Other Options Are Wrong:
2 : 3 : 4 and 3 : 4 : 6 reverse the required scaling; 4 : 3 : 2 inverts the correct order; 12 : 8 : 6 is an unsimplified multiple of the correct ratio.

Common Pitfalls:
Forgetting to clear denominators, or assuming A : B : C equals 2 : 3 : 4 directly from 2A = 3B = 4C, which is incorrect because the coefficients multiply the variables.

Final Answer:
6 : 4 : 3