A and B are partners in a business. A contributes 1/4 of the capital for 15 months and B received 2/3 of the profit. For how long B's money was used ?

Options

A. A) 3 months
B. B) 6 months
C. C) 10 months
D. D) 12 months

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Contributions
A contributes \( \frac{1}{4} \) of the total capital for 15 months. Let’s denote the total capital as \( x \). Therefore, A's contribution is \( \frac{x}{4} \).

### Step 2: Determine B's Contribution
Since A contributes \( \frac{1}{4} \) of the capital, B must contribute the remaining \( \frac{3}{4} \) of the capital. Thus, B's contribution is \( \frac{3x}{4} \).

### Step 3: Profit Sharing
B receives \( \frac{2}{3} \) of the total profit. This means A receives the remaining \( \frac{1}{3} \) of the profit.

### Step 4: Set Up the Ratio of Contributions
The profit is shared based on the product of the capital contributed and the time for which it was invested.

- A's investment duration: \( \frac{x}{4} \) for 15 months.
- B's investment duration: \( \frac{3x}{4} \) for \( y \) months.

### Step 5: Write the Profit Ratio
The profit ratio can be expressed as:
\[ \text{A's share of profit} : \text{B's share of profit} = \left(\frac{x}{4} \times 15\right) : \left(\frac{3x}{4} \times y\right) \]
This simplifies to:
\[ \frac{15}{4} : \frac{3y}{4} \]
Removing the common factor \( \frac{1}{4} \), we have:
\[ 15 : 3y \]

### Step 6: Set Up the Equation
Since we know B's share of the profit is \( \frac{2}{3} \) and A's share is \( \frac{1}{3} \), we can equate the ratios:
\[ \frac{1}{3} : \frac{2}{3} = 15 : 3y \]

### Step 7: Cross Multiply
Cross-multiplying gives us:
\[ 1 \cdot 3y = 2 \cdot 15 \]
This simplifies to:
\[ 3y = 30 \]

### Step 8: Solve for \( y \)
Dividing both sides by 3 gives:
\[ y = 10 \]

### Conclusion
B's money was used for **10 months**.

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