A, B and C together start a business. Three times the investment of A is equal to four times the investment of B, and the capital of B is twice the capital of C. What is the ratio of their shares in the profit?

Introduction / Context:
This partnership question focuses on expressing several capital relationships in a simple ratio form. When time of investment is the same for all partners, profit is directly proportional to the invested capital. By translating the given equalities between the capitals of A, B and C into a ratio, we can find the ratio of their shares in the profit without any explicit profit value being given.

Given Data / Assumptions:
Three times the investment of A equals four times the investment of B. Capital of B is twice the capital of C. All three invest for the same duration. Profit shares are directly proportional to their capitals.

Concept / Approach:
We assign variables to capitals of A, B and C and then use the given equations to relate them. Solving these equations gives each capital in terms of a single variable, leading to a clean ratio. Since profit shares are proportional to capitals, the final capital ratio is the same as the profit ratio we are asked to find. This is a standard technique for handling chained ratio conditions.

Step-by-Step Solution:
Step 1: Let investments be A, B and C for the three partners. Step 2: We are told that 3A = 4B. Step 3: We are also told that B = 2C. Step 4: From B = 2C, write C = B / 2. Step 5: From 3A = 4B, write A = 4B / 3. Step 6: Now express A, B and C all in terms of C for easier comparison. Step 7: Since B = 2C, substitute in A = 4B / 3 to get A = 4 * (2C) / 3 = 8C / 3. Step 8: So A : B : C = 8C / 3 : 2C : C. Step 9: Multiply all terms by 3 to clear the denominator: 8C : 6C : 3C. Step 10: Cancel C to get capital ratio A : B : C = 8 : 6 : 3. Step 11: Because time is equal, profit ratio is also 8 : 6 : 3.

Verification / Alternative check:
We can pick specific numerical values consistent with this ratio. For example, let C = 3 units, then B = 2C = 6 units and A must satisfy 3A = 4B = 24, so A = 8 units. This gives A : B : C = 8 : 6 : 3 which matches the derived ratio. Any scaling of these numbers preserves all given conditions, confirming that 8 : 6 : 3 is the correct and unique ratio for their capitals and profits.

Why Other Options Are Wrong:
Ratios 8 : 3 : 6, 3 : 8 : 6 and 3 : 6 : 8 do not satisfy the relationships 3A = 4B and B = 2C simultaneously. Checking them quickly shows that they fail either the multiple condition between A and B or the relation between B and C. Only the ratio 8 : 6 : 3 preserves both constraints and therefore is the only possible profit ratio.

Common Pitfalls:
A common mistake is to misinterpret the order of partners in the ratio or to incorrectly manipulate the equations, for example writing 3A / 4B = 1 and then guessing values. Another issue is forgetting to clear denominators properly, which leads to fractional ratios that are not simplified. Stepwise substitution and scaling as done above avoids confusion and ensures a consistent ratio.

Final Answer:
The ratio of their shares in the profit is 8 : 6 : 3 for A, B and C respectively.