P and Q invested some money in a business in the ratio of 4 : 7 respectively. After 6 months P withdraws his capital. If they receive profits in the ratio of 4 :21, then for how many months did Q invest in the business?

Options

A. 12
B. 15
C. 18
D. 12

Answer

To solve the problem step by step, we will follow the given information and derive the solution systematically.

### Step 1: Understand the Investment Ratios
P and Q invested in a business in the ratio of 4:7.
- Let P's investment be \(4x\) and Q's investment be \(7x\), where \(x\) is a constant.

### Step 2: Determine the Time of Investment
P invests his capital for 6 months, and we need to find out for how many months Q invested his capital. Let’s denote the number of months Q invested as \(T\).

### Step 3: Set Up the Profit Ratio
The profit received by P and Q is in the ratio of 4:21. According to the profit-sharing rule, the profit is proportional to the product of the amount invested and the time for which it is invested.

### Step 4: Write the Equation for Profit
The profit ratio can be expressed as:
\[ \frac{\text{Profit of P}}{\text{Profit of Q}} = \frac{4x \cdot 6}{7x \cdot T} \]
This simplifies to:
\[ \frac{24x}{7xT} = \frac{4}{21} \]
Here, \(x\) cancels out from both sides.

### Step 5: Cross-Multiply to Solve for T
Cross-multiplying gives us:
\[ 24 \cdot 21 = 4 \cdot 7T \]
This simplifies to:
\[ 504 = 28T \]

### Step 6: Solve for T
Now, divide both sides by 28:
\[ T = \frac{504}{28} = 18 \]

### Conclusion
Thus, Q invested in the business for **18 months**.