Relating capitals across three partners: P’s capital equals twice Q’s capital, and Q’s capital equals three times R’s capital. Determine the capital ratio P : Q : R.

Introduction / Context:
Capital relations can be chained to express all amounts in terms of a single base variable. Then we immediately read the ratio after simplification. This is a standard proportion-conversion exercise used in partnership and ratio topics.

Given Data / Assumptions:

• P = 2Q.
• Q = 3R.

Concept / Approach:
Substitute sequentially: since Q = 3R, then P = 2Q = 2 * 3R = 6R. Now write P : Q : R in terms of R, and simplify to integers if needed (already integers here).

Step-by-Step Solution:

Verification / Alternative check:
Check relations: P is twice Q (6 : 3), and Q is thrice R (3 : 1). Both are satisfied by 6 : 3 : 1.

Why Other Options Are Wrong:
Other options do not maintain both conditions P = 2Q and Q = 3R simultaneously; some invert or misplace multipliers.

Common Pitfalls:
Mixing “times” with “plus” (additive) relations or failing to substitute stepwise; always express all variables via one base before forming the ratio.

Final Answer:
6 : 3 : 1