To solve this problem, we need to determine the share of profit for B based on the investment ratios of A, B, and C. Let's break down the problem step-by-step:

Step 1: Determine the Ratios of Investments

1. Ratio of investments between A and C is given as $2:1$. Let the investments of A and C be $2x$ and $x$, respectively.
2. Ratio of investments between A and B is given as $3:2$. Let the investments of A and B be $3y$ and $2y$, respectively.

Step 2: Find the Common Multiple for A's Investment

Since A's investment needs to be consistent in both ratios, we set $2x = 3y$. This will allow us to express all investments in terms of one variable.

From $2x = 3y$, we can express $x$ in terms of $y$:

$x = \frac{3y}{2}$

Step 3: Express Investments of A, B, and C in Terms of $y$

• A's investment: $2x = 3y$ (by given ratio $3:2$ between A and B)
• B's investment: $2y$
• C's investment: $x = \frac{3y}{2}$

Now, let's put all investments in terms of $y$:

• A's investment: $3y$
• B's investment: $2y$
• C's investment: $\frac{3y}{2}$

Step 4: Calculate the Total Investment

$\text{Total Investment} = A + B + C = 3y + 2y + \frac{3y}{2}$

$= 5y + \frac{3y}{2}$

$= \frac{10y}{2} + \frac{3y}{2}$

$= \frac{13y}{2}$

Step 5: Calculate B's Share in the Total Profit

The total profit is Rs. 1,57,300. We need to find B's share of this profit.

The share of profit is based on the investment ratio. B's investment is $2y$ and the total investment is $\frac{13y}{2}$.

$\text{B's share of profit} = \left( \frac{\text{B's investment}}{\text{Total investment}} \right) \times \text{Total profit}$

$= \left( \frac{2y}{\frac{13y}{2}} \right) \times 1,57,300$

$= \left( \frac{2y \times 2}{13y} \right) \times 1,57,300$

$= \left( \frac{4}{13} \right) \times 1,57,300$

Step 6: Calculate the Final Amount

$= \frac{4 \times 1,57,300}{13}$

$= \frac{6,29,200}{13}$

$= 48,400$

Conclusion

B received Rs. 48,400 from the total profit.

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Would you like more details or have any questions?

Here are some related questions for further understanding:

1. How would the profit share change if the total profit was different?
2. What if the ratio of investment between A and B was $4:3$ instead of $3:2$?
3. How do changes in investment ratios affect the distribution of profit?
4. What would happen if another partner, D, joined the business with a different investment ratio?
5. Can this method be applied if profits are not distributed in proportion to investments?

Tip: Understanding ratios and how they are used to distribute profits is essential in business partnerships.