Business Investment Profit Sharing Calculation

This problem involves calculating the share of profit for one of the partners in a business based on their initial investment. When multiple individuals invest in a business together, the profit (or loss) generated is typically divided among them according to the ratio of their investments.

Step 1: Determine the Investment Ratio

First, let's list the investments made by each friend:

• Friend A invested: $₹6,000$
• Friend B invested: $₹8,000$
• Friend C invested: $₹10,000$

The ratio of their investments is therefore:

$ \text{A : B : C} = 6000 : 8000 : 10000 $

Step 2: Simplify the Investment Ratio

To make the calculations easier, we simplify this ratio by dividing each part by their greatest common divisor, which is $2000$:

$ \text{Simplified Ratio} = \frac{6000}{2000} : \frac{8000}{2000} : \frac{10000}{2000} $

This gives us:

$ \text{A : B : C} = 3 : 4 : 5 $

Step 3: Calculate the Total Parts of the Ratio

The total number of parts in the ratio represents the whole profit. We sum the simplified ratio parts:

$ \text{Total Ratio Parts} = 3 + 4 + 5 = 12 $

Step 4: Calculate C's Share of the Profit

The total profit at the end of the year is given as $₹12,000$. C's share of the profit is calculated based on their portion of the total investment ratio.

The formula for C's share is:

$ \text{C's Share} = \left( \frac{\text{C's Ratio Part}}{\text{Total Ratio Parts}} \right) \times \text{Total Profit} $

Substituting the values we found:

$ \text{C's Share} = \left( \frac{5}{12} \right) \times ₹12,000 $

Now, let's perform the calculation:

$ \text{C's Share} = 5 \times \left( \frac{₹12,000}{12} \right) $

$ \text{C's Share} = 5 \times ₹1,000 $

$ \text{C's Share} = ₹5,000 $

Conclusion

Therefore, C's share of the total profit is $₹5,000$.