A and B entered into a partnership and invested capital in the ratio of `4 :5`. At the end of 8 months A withdraws half of his amount. If at the end of one year, they receive profits.in the ratio of`4 :5`, then B’s capital was used for how many months?

Options

A. 8 months
B. 5 months
C. 11 months
D. 10 months

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Capital Investment
A and B invested capital in the ratio of 4:5. Let's assume A's investment is \(4x\) and B's investment is \(5x\).

### Step 2: Calculate A's Investment for the First 8 Months
A's capital is \(4x\) and he invests this for the first 8 months. The total investment by A for 8 months is:
\[ \text{Investment by A for 8 months} = 4x \times 8 = 32x \]

### Step 3: A Withdraws Half of His Capital
After 8 months, A withdraws half of his capital. Therefore, A's remaining capital becomes:
\[ \text{Remaining capital of A} = \frac{4x}{2} = 2x \]
A will now invest \(2x\) for the remaining 4 months (since the total time is 12 months).

### Step 4: Calculate A's Investment for the Remaining 4 Months
The total investment by A for the remaining 4 months is:
\[ \text{Investment by A for 4 months} = 2x \times 4 = 8x \]

### Step 5: Calculate Total Investment by A
Now, we can calculate the total investment by A over the year:
\[ \text{Total investment by A} = 32x + 8x = 40x \]

### Step 6: Calculate B's Investment
B's capital is \(5x\) and he invests this for the entire year (12 months). Therefore, B's total investment is:
\[ \text{Total investment by B} = 5x \times 12 = 60x \]

### Step 7: Set Up the Ratio of Investments
The profits are distributed in the ratio of 4:5, which corresponds to their investments. Thus, we can set up the equation:
\[ \frac{\text{Total investment by A}}{\text{Total investment by B}} = \frac{4}{5} \]
Substituting the total investments:
\[ \frac{40x}{60x} = \frac{4}{5} \]

### Step 8: Determine B's Capital Duration
Let \(n\) be the number of months B's capital was used. We know:
\[ \text{Total investment by B} = 5x \times n \]
Setting up the equation using the ratio of investments:
\[ \frac{32x + 8x}{5x \times n} = \frac{4}{5} \]
This simplifies to:
\[ \frac{40x}{5xn} = \frac{4}{5} \]
Cross-multiplying gives:
\[ 40 \times 5 = 4 \times 5n \]
\[ 200 = 20n \]
\[ n = 10 \]

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

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