Understanding Partnership Profit Distribution

This problem involves calculating a partner's share of profit in a business partnership. The profit distribution depends not only on the amount invested but also on the duration for which the investment was made. We need to find Chetan's share of the total profit based on his investment and time period relative to Atul and Bharat.

Calculating Investment-Time Products

To determine the profit share, we first calculate the "investment-time product" for each partner. This product represents the effective capital contributed over time.

• Atul's Investment-Time Product: $₹25,000 \times 6 \text{ months} = 150,000$
• Bharat's Investment-Time Product: $₹30,000 \times 8 \text{ months} = 240,000$
• Chetan's Investment-Time Product: $₹40,000 \times 9 \text{ months} = 360,000$

Determining the Profit-Sharing Ratio

The profit is shared in proportion to these investment-time products. We find the ratio of these products:

Ratio (Atul : Bharat : Chetan) = $150,000 : 240,000 : 360,000$

To simplify this ratio, we can divide each part by their greatest common divisor. First, divide by $10,000$:

Ratio = $15 : 24 : 36$

Now, divide by $3$:

Simplified Ratio = $5 : 8 : 12$

Calculating Chetan's Profit Share

The total profit is $₹37,000$. This profit will be divided among the partners according to the ratio $5 : 8 : 12$.

First, find the sum of the ratio parts:

Sum of ratio parts = $5 + 8 + 12 = 25$

Chetan's share corresponds to the '12' part of the ratio. To find his share of the total profit, we use the following formula:

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

Substitute the values:

Chetan's Share = $ \left( \frac{12}{25} \right) \times ₹37,000 $

Now, perform the calculation:

Chetan's Share = $ 12 \times \left( \frac{₹37,000}{25} \right) $

Chetan's Share = $ 12 \times ₹1,480 $

Chetan's Share = $ ₹17,760 $

Conclusion

Based on the calculations, Chetan's share of the total profit is $₹17,760$.