If 7A = 5B = 2C, then what is the ratio A : B : C?

Introduction / Context:
This problem involves three variables A, B and C linked by equalities of the form 7A = 5B = 2C. We are asked to find the ratio A : B : C. Such questions test the ability to introduce a common constant and express each variable in terms of that constant, then extract the required ratio in simplest form.

Given Data / Assumptions:

• 7A = 5B = 2C.
• We must determine A : B : C.
• All variables are assumed to be positive.

Concept / Approach:
When several expressions are equal, we can set each equal to a common constant, say k. Then 7A = k, 5B = k and 2C = k. From this, we can solve for A, B and C in terms of k and then write their ratio. The ratios naturally simplify when we clear denominators and remove any common factors.

Step-by-Step Solution:

Verification / Alternative check:
Choose k = 70 for convenience. Then A = 70/7 = 10, B = 70/5 = 14, C = 70/2 = 35. Check the original relation: 7A = 7 * 10 = 70, 5B = 5 * 14 = 70 and 2C = 2 * 35 = 70. All three match 70, so our values are consistent and the ratio 10 : 14 : 35 is correct.

Why Other Options Are Wrong:

• 35 : 14 : 10 reverses the correct order and would only hold if the variables were assigned differently.
• 14 : 10 : 35 does not follow from the equalities because it would give 7A different from 5B or 2C.
• 2 : 5 : 7 reflects the denominators rather than the actual values of A, B and C.

Common Pitfalls:
Some students incorrectly take 7A, 5B and 2C themselves as the ratio terms (7 : 5 : 2), which is wrong because those are equal, not proportional. The actual ratio is formed from A, B and C, which are inversely proportional to the coefficients 7, 5 and 2 when equated to a common constant.

Final Answer:
The correct ratio is 10 : 14 : 35.