A and B are partners in a business. They invest in the ratio 5:6 at the end of 8 months A withdraws. If they receive profits in the ratio of 5:9, find how long B's investment was used?

Options

A. 12 months
B. 10 months
C. 15 months
D. 14 months

Answer

To solve the problem step by step, we will use the information provided about the investments and the profit-sharing ratio.

### Step 1: Understand the Investment Ratio
A and B invest in the ratio of 5:6. This means if A invests 5 parts, B invests 6 parts.

### Step 2: Determine the Time of Investment for A
A invests for 8 months before withdrawing.

### Step 3: Set Up the Profit Sharing Ratio
The profit-sharing ratio is given as 5:9. This means that for every 14 parts of profit, A receives 5 parts and B receives 9 parts.

### Step 4: Relate Profit to Investment and Time
The profit is proportional to the product of investment and time. Therefore, we can express the profit for A and B as follows:
- Profit of A = Investment of A × Time of A
- Profit of B = Investment of B × Time of B

### Step 5: Write the Equation Based on the Ratios
Using the profit-sharing ratio:
\[ \frac{\text{Profit of A}}{\text{Profit of B}} = \frac{5}{9} \]
Substituting the investments and times:
\[ \frac{5 \times 8}{6 \times t_B} = \frac{5}{9} \]
where \( t_B \) is the time B invested.

### Step 6: Simplify the Equation
Cancelling out the 5 from both sides:
\[ \frac{8}{6 \times t_B} = \frac{1}{9} \]
Cross-multiplying gives:
\[ 8 \times 9 = 6 \times t_B \]
\[ 72 = 6 \times t_B \]

### Step 7: Solve for \( t_B \)
Dividing both sides by 6:
\[ t_B = \frac{72}{6} = 12 \text{ months} \]

### Conclusion
B's investment was used for 12 months.