Some money is divided among A, B, and C such that 5 times A’s share, 3 times B’s share, and 2 times C’s share are all equal. What is the ratio A : B : C?

Introduction / Context:
The statement “5A, 3B, and 2C are equal” defines a proportional relationship. To extract the basic ratio A : B : C, express A, B, and C in terms of the same common constant and then reduce to whole-number parts.

Given Data / Assumptions:

• 5A = 3B = 2C = k (some positive constant).
• Find A : B : C.

Concept / Approach:
From 5A = k ⇒ A = k/5; from 3B = k ⇒ B = k/3; from 2C = k ⇒ C = k/2. The ratio A : B : C is then (1/5) : (1/3) : (1/2). Convert to a common denominator to write whole-number parts.

Step-by-Step Solution:
A = k/5, B = k/3, C = k/2A : B : C = (1/5) : (1/3) : (1/2)Using denominator 30: (1/5, 1/3, 1/2) = (6/30, 10/30, 15/30) ⇒ 6 : 10 : 15

Verification / Alternative check:
Multiply 6 : 10 : 15 by 5, 3, 2 respectively to see equality: 5*6 = 30; 3*10 = 30; 2*15 = 30. Condition satisfied.

Why Other Options Are Wrong:
They are permutations or scaled versions that do not respect 5A = 3B = 2C when checked.

Common Pitfalls:
Thinking A : B : C equals 5 : 3 : 2 directly; remember the multiples of the shares are equal, not the shares themselves.

Final Answer:
6 : 10 : 15