Understanding the Partnership Profit Distribution

This problem involves a partnership where individuals A, B, and C invest capital for certain periods, and their profits are distributed based on their contributions over time. The fundamental principle governing profit distribution in a partnership is that the profit share of each partner is proportional to the product of the capital invested by that partner and the time for which the capital was invested.

Mathematically, this can be represented as:

\(\text{Profit} \propto \text{Capital} \times \text{Time}\)

Therefore, the ratio of the profits of the partners is equal to the ratio of the products of their respective capital investments and investment times.

Let's denote the capital invested by A, B, and C as \(C_A\), \(C_B\), and \(C_C\), and the time for which they invested as \(T_A\), \(T_B\), and \(T_C\).

We are given the following ratios:

• Capital Ratio \(C_A : C_B : C_C = 5 : 7 : 4\)
• Time Ratio \(T_A : T_B : T_C = x : y : z\) (This is what we need to find)
• Profit Ratio \(P_A : P_B : P_C = 45 : 42 : 28\)

According to the partnership principle:

\(P_A : P_B : P_C = (C_A \times T_A) : (C_B \times T_B) : (C_C \times T_C)\)

Substituting the given ratios, we get:

\(45 : 42 : 28 = (5 \times x) : (7 \times y) : (4 \times z)\)

Calculating the Time Ratio x : y : z

We have the relationship \(5x : 7y : 4z = 45 : 42 : 28\). This means that the terms in the ratios are proportional to each other. We can write this as individual proportionalities:

• \(5x \propto 45 \implies x \propto \frac{45}{5} \implies x \propto 9\)
• \(7y \propto 42 \implies y \propto \frac{42}{7} \implies y \propto 6\)
• \(4z \propto 28 \implies z \propto \frac{28}{4} \implies z \propto 7\)

From these proportionalities, we can determine the ratio \(x : y : z\). The ratio of the proportional values is the required time ratio.

\(x : y : z = 9 : 6 : 7\)

Let's verify this by plugging the ratio \(9 : 6 : 7\) back into the capital and time product ratio:

\((5 \times 9) : (7 \times 6) : (4 \times 7) = 45 : 42 : 28\)

This matches the given profit ratio, confirming our calculated time ratio is correct.

The ratio of the timing of investments \(x : y : z\) is therefore \(9 : 6 : 7\).

Final Answer is \(9 : 6 : 7\)

Revision Table: Partnership Ratios

Additional Information on Partnership Problems

Partnership problems often involve scenarios where partners invest different amounts of capital for different durations. Understanding the relationship between capital, time, and profit is crucial for solving these problems.

• Simple Partnership: When all partners invest capital for the same duration, the profit is distributed solely in the ratio of their capital investments. \(P_A : P_B = C_A : C_B\).
• Compound Partnership: When partners invest capital for different durations, the profit is distributed in the ratio of the products of their capital and time. \(P_A : P_B = (C_A \times T_A) : (C_B \times T_B)\). The problem discussed here is an example of a compound partnership.
• Calculations: To find an unknown ratio (Capital, Time, or Profit), you can set up the proportionality as shown in the solution. If you know any two ratios, you can find the third. For example, to find the Time Ratio, divide the Profit Ratio terms by the corresponding Capital Ratio terms.
• Ratio Simplification: Always express the final ratio in its simplest form by dividing all terms by their greatest common divisor (GCD). In this problem, the ratio \(9:6:7\) is already in its simplest form.