Three partners A, B and C start a business together and invest their capitals in the ratio 8 : 5 : 3. At the end of the business period the ratio of their profits is 4 : 15 : 6. What is the ratio of the time periods for which A, B and C invested their money in the business?

Introduction / Context:
This problem involves partnership, where profit sharing depends on both the amount of capital invested and the duration of investment. The question gives the ratio of capitals and the ratio of final profits and asks us to determine the ratio of the time periods for which each partner kept their money in the business. This is a classic concept in partnership questions seen in aptitude exams.

Given Data / Assumptions:

• Capital ratio of A : B : C = 8 : 5 : 3.
• Profit ratio of A : B : C = 4 : 15 : 6.
• Let the time periods of investment for A, B and C be t1, t2 and t3 respectively.
• We assume that profit is directly proportional to (capital * time).

Concept / Approach:
In partnership problems, the share of profit for each partner is proportional to the product of their capital and investment time. In symbolic form, Profit ∝ Capital * Time. Therefore, for each partner, Capital * Time is proportional to the given profit ratio values. We can set equations based on these proportionalities and solve for the relative times. After that we convert them to the simplest integer ratio form.

Step-by-Step Solution:
Let the time periods be t1, t2, t3 for A, B and C. Then 8 * t1 : 5 * t2 : 3 * t3 = 4 : 15 : 6. This means 8 * t1 / 4 = 5 * t2 / 15 = 3 * t3 / 6. Simplify each fraction: 8 * t1 / 4 = 2 * t1; 5 * t2 / 15 = t2 / 3; 3 * t3 / 6 = t3 / 2. So we have 2 * t1 = t2 / 3 = t3 / 2 = k, for some constant k. From 2 * t1 = k, we get t1 = k / 2. From t2 / 3 = k, we get t2 = 3k. From t3 / 2 = k, we get t3 = 2k. Therefore, t1 : t2 : t3 = (k / 2) : 3k : 2k. Multiply all terms by 2 to remove the denominator: t1 : t2 : t3 = 1 : 6 : 4.

Verification / Alternative check:
We can verify by forming the effective capital time products. If we assume k = 2 for convenience, then t1 = 1, t2 = 6 and t3 = 4. Capital products are: A → 8 * 1 = 8, B → 5 * 6 = 30, C → 3 * 4 = 12. The ratio 8 : 30 : 12 simplifies by dividing by 2 to 4 : 15 : 6, which exactly matches the given profit ratio. This confirms that the time ratio 1 : 6 : 4 is consistent.

Why Other Options Are Wrong:
Ratios like 1 : 6 : 5 or 2 : 3 : 4 do not give profit ratios equal to 4 : 15 : 6 when multiplied with the given capital ratio 8 : 5 : 3. The options 2 : 6 : 3 and 3 : 2 : 1 similarly produce different profit distributions and therefore do not satisfy the condition.

Common Pitfalls:
Students commonly confuse the relationship and try to directly equate the profit ratio with the capital ratio, ignoring time. Another frequent mistake is incorrect handling of proportion when solving 8 * t1 : 5 * t2 : 3 * t3 = 4 : 15 : 6. Writing and equating each part systematically and simplifying fractions carefully avoids errors. It is also important to simplify the final ratio completely to its lowest terms.

Final Answer:
The ratio of the time periods for which A, B and C invested their capital is 1 : 6 : 4.