A and B enter into a partnership. A contributed 5000 for 8 months and B, 6000 for 5 months. Find A's share in a total profit of 8400.

Options

A. 4800
B. 2400
C. 3600
D. none of these

Answer

To solve the problem step by step, we will calculate A's share in the total profit based on their contributions and the duration of their investments.

### Step 1: Determine the contributions and time of investments
- A's contribution: \( P_A = 5000 \)
- A's investment duration: \( T_A = 8 \) months
- B's contribution: \( P_B = 6000 \)
- B's investment duration: \( T_B = 5 \) months

### Step 2: Convert months into years
- For A: \( T_A = \frac{8}{12} = \frac{2}{3} \) years
- For B: \( T_B = \frac{5}{12} \) years

### Step 3: Calculate the product of principal and time for both A and B
- A's contribution in terms of time: \( P_A \times T_A = 5000 \times \frac{2}{3} = \frac{10000}{3} \)
- B's contribution in terms of time: \( P_B \times T_B = 6000 \times \frac{5}{12} = 2500 \)

### Step 4: Set up the ratio of their contributions
- The ratio of A's and B's contributions is:
\[ \text{Ratio} = \frac{P_A \times T_A}{P_B \times T_B} = \frac{\frac{10000}{3}}{2500} \]
- Simplifying the ratio:
\[ \frac{10000}{3} \div 2500 = \frac{10000}{3} \times \frac{1}{2500} = \frac{10000}{7500} = \frac{4}{3} \]

### Step 5: Express the profit shares in terms of a variable
- Let A's share of profit be \( 4x \) and B's share be \( 3x \).

### Step 6: Calculate the total profit in terms of x
- Total profit = A's share + B's share:
\[ 4x + 3x = 7x \]
- Given total profit = 8400, we can set up the equation:
\[ 7x = 8400 \]

### Step 7: Solve for x
- Dividing both sides by 7:
\[ x = \frac{8400}{7} = 1200 \]

### Step 8: Calculate A's share of the profit
- A's share:
\[ 4x = 4 \times 1200 = 4800 \]

### Conclusion
A's share in the total profit of 8400 is **4800**.

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