Three partners A, B and C share the profit of a business such that three times the share of A is equal to two times the share of B and also equal to twelve times the share of C. Based on this information, what is the ratio of the profits of A, B and C respectively?

Introduction / Context:
This is a conceptual partnership and ratio problem. Instead of dealing with actual rupee amounts, we are given proportional relationships between the profit shares of three partners A, B and C. Our task is to convert the given proportional statements into a clear numerical ratio of their profit shares.

Given Data / Assumptions:

• Three partners A, B and C share the profit of a business.
• Three times the share of A is equal to two times the share of B.
• The same quantity is also equal to twelve times the share of C.
• We need the ratio of A : B : C.

Concept / Approach:
When multiple expressions are equal, we set them equal to a common variable and then solve for each individual quantity. Here, 3A, 2B and 12C are all equal to the same value. By expressing A, B and C in terms of that common value, we can derive a simple ratio between them. This is a standard technique in ratio and proportion questions in aptitude tests.

Step-by-Step Solution:
Step 1: Let 3A = 2B = 12C = k, where k is a common proportional constant.Step 2: From 3A = k, we get A = k / 3.Step 3: From 2B = k, we get B = k / 2.Step 4: From 12C = k, we get C = k / 12.Step 5: Now the ratio A : B : C is (k / 3) : (k / 2) : (k / 12).Step 6: Cancel k from all terms to get 1/3 : 1/2 : 1/12.Step 7: Convert to whole numbers using common denominator 12: A = 4/12, B = 6/12, C = 1/12.Step 8: Therefore, A : B : C = 4 : 6 : 1.

Verification / Alternative check:
To verify, assume an actual value for k, for example k = 12. Then A = 12 / 3 = 4, B = 12 / 2 = 6, C = 12 / 12 = 1. Check the condition: 3A = 3 * 4 = 12, 2B = 2 * 6 = 12, 12C = 12 * 1 = 12. All are equal, confirming that the ratio 4 : 6 : 1 is consistent with the original condition.

Why Other Options Are Wrong:
3 : 2 : 12 directly uses the multipliers but does not represent the actual shares of A, B and C. 12 : 2 : 3 and 1 : 6 : 4 do not satisfy the condition 3A = 2B = 12C when tested. 'None of these' is incorrect because option 4 : 6 : 1 exactly matches the proportional relationship given in the problem.

Common Pitfalls:
A common error is to misread the condition as A : B : C = 3 : 2 : 12, which is wrong. Another mistake is to forget to express A, B and C separately in terms of a common variable before forming the ratio. Always remember that ratios should reflect the original quantities (A, B and C), not the multipliers applied to them in the condition.

Final Answer:
The correct ratio of the profits of A, B and C is 4 : 6 : 1.