A, B, and C start a business. A invests three times B’s investment, and B invests two-thirds of C’s investment. What is the ratio of their capitals A : B : C?

Introduction / Context:
Translating linked statements into a single three-term ratio is common in partnership problems. We use B as the base to express A and C, then simplify the ratio.

Given Data / Assumptions:

• A = 3 * B.
• B = (2/3) * C ⇒ C = (3/2) * B.

Concept / Approach:
Write A : B : C in terms of B, then clear fractions by multiplying all terms with a common multiple to express the ratio in integers.

Step-by-Step Solution:
A : B : C = 3B : B : 1.5B. Multiply by 2 to remove the decimal: 6B : 2B : 3B. Thus, A : B : C = 6 : 2 : 3.

Verification / Alternative check:
The ratio is consistent with the statements: A = 3B (6 : 2) and B = (2/3)C (2 : 3).

Why Other Options Are Wrong:
3 : 9 : 2 and 6 : 10 : 15 do not follow from the given relationships. 5 : 3 : 2 is unrelated.

Common Pitfalls:
Inverting B and C or mishandling the two-thirds relationship. Carefully convert B = 2/3 of C into C = 3/2 of B first.

Final Answer:
6 : 2 : 3