Harsh Vardhan started a business by investing Rs. 36000. After 4 months Gyan Vardhan joined him with some investment. At the end of the year, the total profit was divided between them in the ratio of 9:7. How much capital was invested by Gyan Vardhan in the business?

Answer

To solve the problem step by step, we will follow the information provided and the relationships between the investments and the profits.

### Step 1: Identify the Investment and Time for Harsh Vardhan
Harsh Vardhan invested Rs. 36,000 and he was in the business for the entire year (12 months).

### Step 2: Identify the Investment and Time for Gyan Vardhan
Gyan Vardhan joined after 4 months, so he was in the business for 12 - 4 = 8 months. Let the amount Gyan Vardhan invested be Rs. x.

### Step 3: Set Up the Profit Ratio
The profits are divided in the ratio of 9:7. We can express the profits in terms of their investments and the time they were invested:
- Profit of Harsh Vardhan (P1) = Investment × Time = 36,000 × 12
- Profit of Gyan Vardhan (P2) = Investment × Time = x × 8

### Step 4: Write the Equation Based on the Profit Ratio
According to the ratio of profits:
\[ \frac{P1}{P2} = \frac{9}{7} \]
Substituting the expressions for P1 and P2:
\[ \frac{36,000 \times 12}{x \times 8} = \frac{9}{7} \]

### Step 5: Cross-Multiply to Solve for x
Cross-multiplying gives:
\[ 7 \times (36,000 \times 12) = 9 \times (x \times 8) \]
Calculating the left side:
\[ 7 \times 36,000 \times 12 = 302,400 \]
So the equation becomes:
\[ 302,400 = 72x \]

### Step 6: Solve for x
Now, divide both sides by 72:
\[ x = \frac{302,400}{72} \]
Calculating this gives:
\[ x = 4,200 \]

### Step 7: Conclusion
Thus, Gyan Vardhan invested Rs. 42,000 in the business.