If A/3 = B/2 = C/5 for three real numbers A, B and C, then what is the ratio (C + A)^2 : (A + B)^2 : (B + C)^2?

Introduction / Context:
This is a ratio based algebra question that checks understanding of proportional relationships and simple substitution. By expressing A, B, and C in terms of a common parameter, we can easily compute the required squared ratio. Such questions are standard in quantitative aptitude sections of many examinations.

Given Data / Assumptions:

A/3 = B/2 = C/5.
A, B, and C are real numbers, and denominators are non zero.
The required ratio is (C + A)^2 : (A + B)^2 : (B + C)^2.

Concept / Approach:
Whenever three fractions are equal, we introduce a common parameter, say k. We then write A = 3k, B = 2k, C = 5k. The expressions involving A, B, and C can then be simplified in terms of k alone. Since any common factor in the numerator and denominator of a ratio cancels out, k will drop out and we will obtain a simple numerical ratio.

Step-by-Step Solution:
Let A/3 = B/2 = C/5 = k.Then A = 3k, B = 2k, C = 5k.Compute C + A = 5k + 3k = 8k, so (C + A)^2 = (8k)^2 = 64k^2.Compute A + B = 3k + 2k = 5k, so (A + B)^2 = (5k)^2 = 25k^2.Compute B + C = 2k + 5k = 7k, so (B + C)^2 = (7k)^2 = 49k^2.Therefore the ratio (C + A)^2 : (A + B)^2 : (B + C)^2 = 64k^2 : 25k^2 : 49k^2.Divide every term by k^2: 64 : 25 : 49.

Verification / Alternative check:
We can choose a convenient value of k, for example k = 1. Then A = 3, B = 2, C = 5. Substitute: (C + A)^2 = (5 + 3)^2 = 8^2 = 64, (A + B)^2 = (3 + 2)^2 = 5^2 = 25, and (B + C)^2 = (2 + 5)^2 = 7^2 = 49. This gives the ratio 64 : 25 : 49, which confirms our algebraic derivation.

Why Other Options Are Wrong:
The other options are permutations of similar numbers but do not match the actual order that comes from the expressions involving A, B, and C. For example, 25 : 4 : 9 ignores the contribution from the coefficients 3, 2, and 5 in the correct combinations. Only 64 : 25 : 49 matches (8k)^2 : (5k)^2 : (7k)^2.

Common Pitfalls:
A common mistake is to misassign the parameter, such as writing A = k/3 instead of A = 3k, which reverses the relationship. Another error is to forget to square the sums, leading to the ratio 8 : 5 : 7 instead of 64 : 25 : 49. Carefully following the substitution and respecting the squares avoids these issues.

Final Answer:
The required ratio (C + A)^2 : (A + B)^2 : (B + C)^2 is 64 : 25 : 49.