Understanding the Business Partnership Problem

This problem involves calculating the initial investments made by business partners A and B, based on their investment durations, C's investment, and specific profit-sharing conditions.

Let's break down the information given:

• Partners and Investment Time:
  • A invested for $T_A = 3$ months.
  • B invested for $T_B = 4$ months.
  • C invested for $T_C = 6$ months.
• C's Investment: $I_C = \text{Rs. } 2400$.
• Profit Share Relationships:
  • A's profit share ($P_A$) is $\frac{2}{3}$ of B's profit share ($P_B$): $P_A = \frac{2}{3} P_B$.
  • C's profit share ($P_C$) is $\frac{1}{2}$ of the total profit ($P_{Total}$): $P_C = \frac{1}{2} P_{Total}$.
• Goal: Find the investments of A ($I_A$) and B ($I_B$).

Calculating the Profit Sharing Ratio

The profit share in a partnership is generally proportional to the product of the investment amount and the duration for which it was invested. That is, Profit $\propto$ Investment $\times$ Time.

So, the ratio of profits is:

$P_A : P_B : P_C = (I_A \times T_A) : (I_B \times T_B) : (I_C \times T_C)$

Substituting the given time periods:

$P_A : P_B : P_C = (I_A \times 3) : (I_B \times 4) : (I_C \times 6)$

Now, let's use the given profit share relationships to find the ratio $P_A : P_B : P_C$ directly.

We know $P_C = \frac{1}{2} P_{Total}$. Since the total profit is the sum of individual profits ($P_{Total} = P_A + P_B + P_C$), this implies:

$P_C = \frac{1}{2} (P_A + P_B + P_C)$

Multiplying by 2:

$2 P_C = P_A + P_B + P_C$

Rearranging the terms gives:

$P_C = P_A + P_B$

We are also given $P_A = \frac{2}{3} P_B$. Substitute this into the equation $P_C = P_A + P_B$:

$P_C = (\frac{2}{3} P_B) + P_B$

$P_C = (\frac{2}{3} + 1) P_B$

$P_C = \frac{5}{3} P_B$

Now we can establish the ratio between the profits:

$P_A : P_B : P_C = \frac{2}{3} P_B : P_B : \frac{5}{3} P_B$

To simplify this ratio, we can divide each term by $P_B$:

$\frac{2}{3} : 1 : \frac{5}{3}$

To get whole numbers, multiply every part of the ratio by 3:

$(\frac{2}{3} \times 3) : (1 \times 3) : (\frac{5}{3} \times 3)$

$2 : 3 : 5$

So, the profit sharing ratio is $P_A : P_B : P_C = 2 : 3 : 5$.

Determining the Investment Amounts

We have established two ways to represent the profit ratios:

1. Based on investment and time: $P_A : P_B : P_C = 3I_A : 4I_B : 6I_C$
2. Derived from profit share conditions: $P_A : P_B : P_C = 2 : 3 : 5$

Let's equate these ratios:

$3I_A : 4I_B : 6I_C = 2 : 3 : 5$

We know $I_C = 2400$ and $T_C = 6$. The investment-time product for C is:

$I_C \times T_C = 2400 \times 6 = 14400$

Now, we can relate the parts of the ratio corresponding to C:

$6I_C$ in the first ratio corresponds to $5$ in the second ratio.

So, $14400$ corresponds to $5$ parts of the profit ratio.

Let's find the value of one part:

Value of 1 part $= \frac{14400}{5} = 2880$

Now we can find the values corresponding to A and B in the investment-time product ratio:

For A: $3I_A$ corresponds to $2$ parts.

$3I_A = 2 \times (\text{Value of 1 part}) = 2 \times 2880 = 5760$

For B: $4I_B$ corresponds to $3$ parts.

$4I_B = 3 \times (\text{Value of 1 part}) = 3 \times 2880 = 8640$

Finally, calculate the investments $I_A$ and $I_B$:

Investment of A ($I_A$):

$3I_A = 5760 \implies I_A = \frac{5760}{3} = 1920$

Investment of B ($I_B$):

$4I_B = 8640 \implies I_B = \frac{8640}{4} = 2160$

Therefore, A invested Rs. 1920 and B invested Rs. 2160.