Question

A and B started a partnership business investing some amount in the ratio of 3 : 5. C joined them after six months with an amount equal to that B. In what proportion should the profit at the end of one year be distributed among A, B and C?

• A. 3 : 5 : 2
• B. 3 : 5 : 5
• C. 6 : 10 : 5
• D. 3 : 7 : 8
• E. 6 : 10 : 3

Hint:

Always multiply the investment ratio by the time duration ratio to get the profit ratio. Be careful to calculate the time the money remained in the business, not the time passed before joining.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This is a partnership problem where the profit distribution depends on two factors: the amount of capital invested and the duration for which it was invested.
We need to calculate the effective investment of A, B, and C over a period of one year (12 months).
Step 2: Key Formula or Approach:
The ratio of profit distribution is equal to the ratio of the product of (Investment \(\times\) Time).
\[\text{Profit Ratio} = (I_A \times T_A) : (I_B \times T_B) : (I_C \times T_C)\]
Step 3: Detailed Explanation:

Assigning Investment Values:
- Let the investment of A be \(3x\).
- Let the investment of B be \(5x\).
- C invests an amount equal to B, so C's investment is also \(5x\).

Determining Time Periods:
- A and B started the business and stayed for the full year. So, \(T_A = 12\) months and \(T_B = 12\) months.
- C joined after 6 months. In a 12-month year, C's money was invested for \(12 - 6 = 6\) months. So, \(T_C = 6\) months.

Calculating Profit Ratios:
- A's share \(\propto 3x \times 12 = 36x\)
- B's share \(\propto 5x \times 12 = 60x\)
- C's share \(\propto 5x \times 6 = 30x\)

Simplifying the Ratio:
- The ratio is \(36x : 60x : 30x\).
- Dividing by \(x\): \(36 : 60 : 30\).
- Dividing by the common factor 6:
\(36 \div 6 = 6\)
\(60 \div 6 = 10\)
\(30 \div 6 = 5\)
- The final ratio is \(6 : 10 : 5\).

Step 4: Final Answer:
The profit should be distributed among A, B, and C in the ratio \(6 : 10 : 5\).