If 6A = 7B = 5C, what is the ratio A : B : C implied by this relationship?

Introduction / Context:
This problem is similar to other chained proportional equations such as 3A = 7B = 13C. It examines whether the student can convert the expression 6A = 7B = 5C into a meaningful ratio for A, B and C. Questions of this type are foundational for ratio and proportion and often appear in aptitude tests and entrance examinations.

Given Data / Assumptions:

• We are given 6A = 7B = 5C.
• A, B and C are positive quantities.
• The task is to find the ratio A : B : C.

Concept / Approach:
Whenever we have 6A = 7B = 5C, we introduce a constant k such that 6A = 7B = 5C = k. From this, we express A, B and C in terms of k, giving A = k / 6, B = k / 7 and C = k / 5. This leads to a ratio in terms of fractions. We then remove denominators by multiplying through by the least common multiple of 6, 7 and 5. The resulting integer ratio is our final answer.

Step-by-Step Solution:
Assume 6A = 7B = 5C = k.Then A = k / 6, B = k / 7 and C = k / 5.So A : B : C = (k / 6) : (k / 7) : (k / 5).Cancel the common factor k to obtain 1 / 6 : 1 / 7 : 1 / 5.Find the least common multiple of 6, 7 and 5, which is 210.Multiply each term by 210: (1 / 6) * 210 = 35, (1 / 7) * 210 = 30, (1 / 5) * 210 = 42.Therefore, A : B : C = 35 : 30 : 42.

Verification / Alternative check:
Take A = 35, B = 30 and C = 42.Compute 6A = 6 * 35 = 210, 7B = 7 * 30 = 210 and 5C = 5 * 42 = 210.All three expressions are equal, which confirms that 6A = 7B = 5C is satisfied by A : B : C = 35 : 30 : 42.

Why Other Options Are Wrong:
Options such as 42 : 30 : 35 or 30 : 35 : 42 represent permutations of the correct ratio but do not preserve the condition 6A = 7B = 5C when checked. The option 5 : 7 : 6 confuses coefficients with the actual ratio and ignores the reciprocal transformation. Therefore, only 35 : 30 : 42 is consistent with the original equality.

Common Pitfalls:
Students sometimes mistakenly take the direct ratio of A : B : C as 6 : 7 : 5, which is incorrect because those are the multipliers of A, B and C, not the variables themselves. Another mistake is to forget cancelling the common constant k before clearing denominators. Always convert the chained equality into fractional expressions, then clear denominators systematically.

Final Answer:
The required ratio is 35 : 30 : 42.