A, B and C are three partners in a business. A, whose money has been used for 4 months, Claims 1/8 of the profit, B whose money has been used for 6 months, claims 1/3 of the profit. C had invested Rs. 1560 for 8 months. How much money did A and B contribute?

Answer

To solve the problem step by step, we will determine the contributions of partners A and B based on the information provided.

### Step 1: Understand the Profit Distribution
A claims \( \frac{1}{8} \) of the total profit \( P \), and B claims \( \frac{1}{3} \) of the total profit \( P \). C's profit can be calculated by subtracting A's and B's claims from the total profit.

### Step 2: Calculate C's Profit
The profit claimed by A and B can be expressed as:
- Profit of A = \( \frac{1}{8} P \)
- Profit of B = \( \frac{1}{3} P \)

To combine these fractions, we need a common denominator. The least common multiple of 8 and 3 is 24.

Convert the fractions:
- Profit of A = \( \frac{1}{8} P = \frac{3}{24} P \)
- Profit of B = \( \frac{1}{3} P = \frac{8}{24} P \)

Now, we can find C's profit:
\[ \text{Profit of C} = P - \left( \frac{3}{24} P + \frac{8}{24} P \right) = P - \frac{11}{24} P = \frac{13}{24} P \]

### Step 3: Set Up the Equation for C's Investment
C invested Rs. 1560 for 8 months. The profit of C can also be calculated as:
\[ \text{Profit of C} = \text{Investment of C} \times \text{Time of C} = 1560 \times 8 = 12480 \]

### Step 4: Equate and Solve for Total Profit \( P \)
From the previous step, we have:
\[ \frac{13}{24} P = 12480 \]

To find \( P \), we rearrange the equation:
\[ P = 12480 \times \frac{24}{13} \]

Calculating this gives:
\[ P = 12480 \times \frac{24}{13} = 12480 \times 1.846153846 = 23040 \]

### Step 5: Calculate A's Investment
We know that A's profit is \( \frac{1}{8} P \):
\[ \text{Profit of A} = \frac{1}{8} \times 23040 = 2880 \]

A's investment can be calculated using the formula:
\[ \text{Investment of A} \times \text{Time of A} = \text{Profit of A} \]
\[ \text{Investment of A} \times 4 = 2880 \]
\[ \text{Investment of A} = \frac{2880}{4} = 720 \]

### Step 6: Calculate B's Investment
Similarly, for B:
\[ \text{Profit of B} = \frac{1}{3} P = \frac{1}{3} \times 23040 = 7680 \]

B's investment is calculated as:
\[ \text{Investment of B} \times \text{Time of B} = \text{Profit of B} \]
\[ \text{Investment of B} \times 6 = 7680 \]
\[ \text{Investment of B} = \frac{7680}{6} = 1280 \]

### Final Contributions
- Investment of A = Rs. 720
- Investment of B = Rs. 1280

### Summary
A contributed Rs. 720 and B contributed Rs. 1280.

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