Understanding Partnership Ratios: Capital, Profit, and Time

In a business partnership, the profit received by each partner is generally proportional to the amount of capital invested and the duration for which the capital was invested. This relationship can be expressed as:

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

Or, we can say that Profit is directly proportional to the product of Capital and Time. Mathematically, this can be written as:

\( P = k \times C \times T \)

where \(P\) is the Profit, \(C\) is the Capital, \(T\) is the Time, and \(k\) is a constant of proportionality.

From this relationship, if we want to find the ratio of the invested times, we can rearrange the formula:

\( T \propto \frac{P}{C} \)

So, the ratio of the invested times of partners A, B, and C is given by the ratio of their respective Profit-to-Capital ratios:

\( T_A : T_B : T_C = \frac{P_A}{C_A} : \frac{P_B}{C_B} : \frac{P_C}{C_C} \)

Applying Given Ratios

We are given the following ratios:

• Ratio of Capital (A : B : C) = \( C_A : C_B : C_C = 3 : 4 : 8 \)
• Ratio of Profit (A : B : C) = \( P_A : P_B : P_C = 2 : 3 : 5 \)

Using the derived relationship \( T_A : T_B : T_C = \frac{P_A}{C_A} : \frac{P_B}{C_B} : \frac{P_C}{C_C} \), we substitute the values from the given ratios:

\( T_A : T_B : T_C = \frac{2}{3} : \frac{3}{4} : \frac{5}{8} \)

Simplifying the Time Ratio

To express this ratio of fractions as a ratio of whole numbers, we need to multiply each fraction by the least common multiple (LCM) of the denominators (3, 4, and 8). The LCM of 3, 4, and 8 is 24.

Multiply each term in the ratio by 24:

\( T_A : T_B : T_C = \left(\frac{2}{3} \times 24\right) : \left(\frac{3}{4} \times 24\right) : \left(\frac{5}{8} \times 24\right) \)

Perform the multiplication:

• For A: \( \frac{2}{3} \times 24 = 2 \times 8 = 16 \)
• For B: \( \frac{3}{4} \times 24 = 3 \times 6 = 18 \)
• For C: \( \frac{5}{8} \times 24 = 5 \times 3 = 15 \)

So, the ratio of their invested time is:

\( T_A : T_B : T_C = 16 : 18 : 15 \)

Conclusion

The ratio of the invested time for partners A, B, and C is 16 ∶ 18 ∶ 15.

Revision Table: Capital, Profit, Time Ratios

Additional Information: Partnership Calculations

Partnership questions often involve calculations based on capital, time, and profit sharing. Key concepts include:

• Profit Sharing: Profits are typically shared in the ratio of (Capital x Time) for each partner.
• Capital Ratio: The ratio of the amounts invested by each partner.
• Time Ratio: The ratio of the duration for which each partner's capital was invested. If all partners invest for the same duration, profit is shared solely in the capital ratio. If partners invest different amounts for different durations, both capital and time are considered.
• Calculating Unknowns: If you are given two of the three components (Capital ratio, Time ratio, Profit ratio), you can find the third using the relationship \( P \propto C \times T \). For example, \( C \propto P/T \) or \( T \propto P/C \).
• LCM for Ratio Simplification: When dealing with ratios involving fractions, find the LCM of the denominators to convert the ratio into its simplest form consisting of whole numbers.

Understanding the direct proportionality between profit and the product of capital and time is fundamental to solving such partnership problems.