P, Q, and R start a business. P invests three times as much as Q, and Q invests two-thirds as much as R. Find the ratio of their capitals P : Q : R.

Introduction / Context:
Partnership setups often give chained relationships between capitals. Translate the verbal conditions into algebra and then reduce to a clean integer ratio.

Given Data / Assumptions:

• P = 3 × Q
• Q = (2/3) × R
• All capitals are positive.

Concept / Approach:
Express P and Q in terms of R, then form P : Q : R and scale to remove fractions. Multiplying all terms by the least common multiple of denominators yields an integer ratio.

Step-by-Step Solution:
Given Q = (2/3)RThen P = 3Q = 3*(2/3)R = 2RP : Q : R = 2R : (2/3)R : RCancel R and clear denominator by ×3 ⇒ 6 : 2 : 3

Verification / Alternative check:
Check back: If R = 3, then Q = 2, P = 6; indeed P is thrice Q, and Q is two-thirds of R.

Why Other Options Are Wrong:
Other orders or numbers violate either “P is 3 times Q” or “Q is 2/3 of R.”

Common Pitfalls:
Confusing “Q is 2/3 of R” with “R is 2/3 of Q.” Read the direction carefully.

Final Answer:
6 : 2 : 3