Profit Distribution in a Business Partnership: A Detailed Analysis Using Algebra and Ratio

Much more than just a high school problem, the scenario described in this article illustrates fundamental principles of profit distribution in a business context. The specifics involve a partnership between two individuals, Sachin and Sumit, and the intricacies of profit allocation based on investment size and duration. This detailed look into the mathematical model will serve as a practical guide for anyone looking to understand such financial distributions.

Introduction

The given problem involves two partners, Sachin and Sumit, with differing investments and time spent in the business. By breaking down the problem into manageable parts, we can utilize algebraic equations to determine the total profit and the proportional share for each partner. This article not only offers a step-by-step explanation but also provides a clear understanding of the methods utilized.

The Scenario

Let's re-examine the problem with the given details:

Sachin started a business with an investment that is 1/3rd the investment of Sumit. The business duration was 2 years, or 24 months. Sumit left after 1/4th of the business duration. Sachin received a profit of Rs. 13,440 at the end of the business period (24 months).

The goal is to determine the total profit for the business.

The Calculation

To simplify, let's denote the total profit by 'P' and use the following notation for investments and durations:

Let Sumit's investment be X. Hence, Sachin's investment is X/3. The total business duration is 24 months. Sumit stayed for 6 months. Sachin stayed for the entire 24 months.

Formulating the Problem

The profit share of each partner can be determined using their respective investment duration ratios. Let's express Sachin's investment duration ratio and calculate his share of the profit.

Sachin's share of the investment over the business period is:

$$ text{Sachin's Share} frac{text{Sachin's Investment} times text{Sachin's Tenure}}{text{Sumit's Investment} times text{Sumit's Tenure} text{Sachin's Investment} times text{Sachin's Tenure}} $$

Substituting the values:

$$ text{Sachin's Share} frac{frac{X}{3} times 24}{X times 6 frac{X}{3} times 24} $$ $$ frac{8X}{8X 8X} $$ $$ frac{8X}{16X} $$ $$ frac{8}{16} $$ $$ frac{1}{2} times frac{8}{14} $$ $$ frac{8}{14} times frac{1}{7} times P $$ $$ frac{8P}{98} $$ $$ frac{4P}{49} times 2 $$ $$ frac{8P}{14} $$ $$ frac{8}{14} times frac{13440}{8} $$ $$ 13440 times frac{14}{8} $$ $$ 23520 $$ $$ P 23520 $$

Thus, the total profit is Rs. 23,520.

Alternative Solutions

Let's validate the solution using alternative methods to ensure the result is consistent:

Method 2

Let Sumit's investment be X.

Sachin's investment is X/3.

Sumit's tenure is 6 months and Sachin's tenure is 24 months.

Total investment x duration for Sumit is 6X and for Sachin is 24X/3 8X.

Combining the total investment for the business is 6X 8X 14X.

Sachin's share of the profit is 8X / 14X 8/14 of the total profit.

Given Sachin's profit is Rs. 13,440, the total profit P can be calculated as:

$$ frac{8P}{14} 13440 $$ $$ P 13440 times frac{14}{8} $$ $$ P 23520 $$

Method 3

Let Sumit's investment be 3K and Sachin's investment be K.

Sumit's tenure is 6 months and Sachin's is 24 months.

Sumit's investment duration is 3K6 18K and Sachin's is K24 24K.

The ratio of their profits is 18K:24K which simplifies to 3:4.

Sachin's profit is Rs. 13,440, corresponding to 4/7 of the total profit P.

Solving for P:

$$ frac{4P}{7} 13440 $$ $$ P 13440 times frac{7}{4} $$ $$ P 3360 times 7 $$ $$ P 23520 $$

Conclusion

This comprehensive solution demonstrates the use of algebraic equations and ratios to solve business partnership problems. It shows that Sachin's total profit is Rs. 13,440 and the total profit of the business is Rs. 23,520. Understanding these methods can help in accurately assessing profit distribution in various business scenarios.