Three partners S, T and U start a business together. Their capitals are in the ratio 3 : 4 : 6. At the end, the ratio of their shares in profit is 1 : 2 : 3. What is the respective ratio of the time periods for which S, T and U invested their capitals?

Introduction / Context:
This partnership problem again focuses on how profit sharing depends on both capital and time. We are given the ratio of capital contributions by S, T and U, and the final ratio of their profits. Using the relationship Profit ∝ Capital * Time, we can determine how long each partner kept their money invested relative to the others. This type of question is typical in quantitative aptitude sections of exams.

Given Data / Assumptions:

• Capital ratio S : T : U = 3 : 4 : 6.
• Profit ratio S : T : U = 1 : 2 : 3.
• Let the time periods of investment be tS, tT and tU.
• Profit for each partner is directly proportional to (capital * time).

Concept / Approach:
If capital and time are known in ratio form for partners in a business, the profit share is proportional to the product of capital and time. Here we know the capital ratio and the resulting profit ratio, but not the time ratio. By setting up proportional relationships between capital time products and the observed profit ratios, we can solve for the relative times and simplify them to their lowest integer ratio form.

Step-by-Step Solution:
Let tS, tT, tU be the time periods for S, T and U. Capital time products are: S → 3 * tS, T → 4 * tT, U → 6 * tU. These are proportional to the profit shares 1 : 2 : 3. So 3 * tS : 4 * tT : 6 * tU = 1 : 2 : 3. This implies 3 * tS / 1 = 4 * tT / 2 = 6 * tU / 3. Simplify: 3 * tS / 1 = 3 * tS, 4 * tT / 2 = 2 * tT, 6 * tU / 3 = 2 * tU. Therefore 3 * tS = 2 * tT = 2 * tU = k, for some constant k. From 3 * tS = k, we get tS = k / 3. From 2 * tT = k, we get tT = k / 2. From 2 * tU = k, we get tU = k / 2. So tS : tT : tU = (k / 3) : (k / 2) : (k / 2). Multiply through by 6 to clear denominators: tS : tT : tU = 2 : 3 : 3.

Verification / Alternative check:
Take k = 6 for convenience. Then tS = 2, tT = 3 and tU = 3. Capital time products: S → 3 * 2 = 6, T → 4 * 3 = 12, U → 6 * 3 = 18. The ratio 6 : 12 : 18 simplifies by dividing by 6 to 1 : 2 : 3, which matches the given profit ratio. This confirms that the time ratio 2 : 3 : 3 is correct.

Why Other Options Are Wrong:
Ratios like 3 : 2 : 2 or 4 : 5 : 3 give capital time products that do not simplify to 1 : 2 : 3. Similarly, 2 : 2 : 3 and 1 : 2 : 3 do not reproduce the correct profit distribution when combined with the original capital ratio 3 : 4 : 6.

Common Pitfalls:
A frequent error is to assume that the ratio of profit is the same as the ratio of capital, ignoring time altogether. Another mistake is algebraic: not setting up the fractions correctly when equating 3 * tS : 4 * tT : 6 * tU to 1 : 2 : 3. Some candidates also forget to fully simplify the final ratio. Working methodically with the capital time products and checking with a quick verification prevents these issues.

Final Answer:
The respective ratio of the time periods for which S, T and U invested their capital is 2 : 3 : 3.