If 3A = 7B = 13C, what is the ratio A : B : C based on this proportional relationship?

Introduction / Context:
This problem checks understanding of proportional equations of the form 3A = 7B = 13C. Such expressions are very common in ratio and proportion questions in aptitude exams. The goal is to convert a chained equality into a clean ratio A : B : C by expressing each variable in terms of a common constant and then simplifying the resulting fractions.

Given Data / Assumptions:

• We are given 3A = 7B = 13C.
• We assume A, B and C are positive real quantities.
• We need to find the ratio A : B : C.

Concept / Approach:
Whenever we see an expression like 3A = 7B = 13C, we can assume that each of these is equal to some common constant k. That is, 3A = k, 7B = k and 13C = k. This allows us to write A, B and C individually in terms of k. Once they are written in fractional form, we can remove the denominator by multiplying all terms by the least common multiple and then read off the final ratio.

Step-by-Step Solution:
Assume 3A = 7B = 13C = k.Then A = k / 3, B = k / 7 and C = k / 13.So A : B : C = (k / 3) : (k / 7) : (k / 13).Cancel the common factor k to get 1 / 3 : 1 / 7 : 1 / 13.To remove denominators, multiply each term by the least common multiple of 3, 7 and 13, which is 273.(1 / 3) * 273 = 91, (1 / 7) * 273 = 39, (1 / 13) * 273 = 21.Therefore A : B : C = 91 : 39 : 21.

Verification / Alternative check:
Take A = 91, B = 39 and C = 21. Then 3A = 273, 7B = 273 and 13C = 273, which confirms that 3A = 7B = 13C.Since the condition is satisfied, the ratio 91 : 39 : 21 is correct.

Why Other Options Are Wrong:
Options like 21 : 39 : 91 or 39 : 91 : 21 reverse or permute the terms incorrectly, so they do not satisfy 3A = 7B = 13C when we test them. The option 13 : 7 : 3 directly copies coefficients and ignores the reciprocal nature of ratios when moving from 3A = 7B = 13C to A : B : C. Thus, only 91 : 39 : 21 works properly.

Common Pitfalls:
A very common mistake is to take the ratio as 3 : 7 : 13 or its simple variations. Candidates sometimes forget that the coefficients 3, 7 and 13 are multiplied with A, B and C, so when we isolate A, B and C, we actually get reciprocals 1 / 3, 1 / 7 and 1 / 13. Not handling this reciprocal step correctly leads to an incorrect ratio. Always express each term in terms of a common constant first and then simplify.

Final Answer:
The required ratio is 91 : 39 : 21.