Bharat and Ekta started a business together. Bharat invested for 11 months while Ekta invested for 12 months. Out of the total profit, 20% is given to Bharat as an active partner, and the remaining profit is divided between Bharat and Ekta in proportion to their investments. In the end, Bharat's and Ekta's profit shares are equal. If Bharat's initial investment is Rs. 6,000, what is Ekta's initial investment?

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**Solution:**

1. Let the total profit be Rs. $100x$.

2. Bharat's investment is Rs. 6,000 for 11 months, so his investment-time product is $6000 	imes 11 = 66000$.

3. Let Ekta's investment be Rs. $y$ for 12 months, so her investment-time product is $y 	imes 12 = 12y$.

4. The profit (after giving 20% to Bharat as an active partner) is divided between Bharat and Ekta in the ratio of their investment-time products:
   $$	ext{Ratio} = 66000 : 12y = 5500 : y$$

5. Bharat's total profit share is 20% of total profit plus his share from the remaining 80%:
   $$	ext{Bharat's profit} = 20\% 	imes 100x + 80\% 	imes 100x 	imes \frac{5500}{5500 + y} = 20x + 80x 	imes \frac{5500}{5500 + y}$$

6. Ekta's profit share is the remaining part of the 80%:
   $$80x 	imes \frac{y}{5500 + y}$$

7. Given that Bharat's and Ekta's profit shares are equal:
   $$20x + 80x 	imes \frac{5500}{5500 + y} = 80x 	imes \frac{y}{5500 + y}$$

8. Simplify the equation:
   $$20x + 80x 	imes \frac{5500}{5500 + y} = 80x - 80x 	imes \frac{5500}{5500 + y}$$

$$20x = 80x - 160x 	imes \frac{5500}{5500 + y}$$

$$160x 	imes \frac{5500}{5500 + y} = 60x$$

$$\frac{5500}{5500 + y} = \frac{60}{160} = \frac{3}{8}$$

9. Cross-multiplied:
   $$8 	imes 5500 = 3 	imes (5500 + y)$$

$$44000 = 16500 + 3y$$

$$3y = 44000 - 16500 = 27500$$

$$y = \frac{27500}{3} = 9166.67$$

**Answer:** Rs. 9,166.67

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