Business Contributions and Mathematical Solutions

In the realm of business, understanding and balancing financial contributions among partners is a key aspect of successful collaboration. This article explores a practical scenario involving four individuals who decided to start a business and the mathematical solutions to determine their respective financial contributions based on given ratios and additional details. The process is detailed step-by-step, offering insights that can be invaluable for aspiring entrepreneurs and financial analysts.

Scenario and Information Provided

Four business partners, A, B, C, and D, decided to initiate a business. The contributions of A and B are in the ratio of 3:2, and the contributions of C and D are in the ratio of 5:4. Additionally, C contributed Sh. 3,000 more than A, and the total amount contributed is Sh. 47,000.

Mathematical Solution

Given the requirements, we can establish the following equations based on the provided information:

The ratio of A's and B's contributions: 3:2 The ratio of C's and D's contributions: 5:4 C contributed Sh. 3,000 more than A Total contributions: Sh. 47,000

To find the individual contributions, we follow these steps:

Step 1: Establishing the Variables

Let A's contribution be represented by ( x )t and B's contribution be ( y ) in Sh. Based on the ratio 3:2, we can write:

[ 3:2 x:y Rightarrow y frac{2}{3}x ]

Step 2: Expressing C's Contribution

C's contribution is 3,000 more than A, so:

[ C A 3000 Rightarrow C x 3000 ]

Step 3: Considering the Ratio of C and D

The ratio of C and D's contributions is 5:4, so if D's contribution is ( z ), we can express:

[ 5:4 C:z Rightarrow z frac{4}{5}C Rightarrow z frac{4}{5}(x 3000) ]

Step 4: Total Contributions Equation

The total contributions are Sh. 47,000, so we set up the equation:

[ x frac{2}{3}x (x 3000) frac{4}{5}(x 3000) 47000 ]

Step 5: Simplifying the Equation

Combining like terms, we get:

[ x frac{2}{3}x x 3000 frac{4}{5}x 2400 47000 ]

[ frac{15}{15}x frac{10}{15}x frac{15}{15}x frac{12}{15}x 5400 47000 ]

[ frac{52}{15}x 5400 47000 ]

[ frac{52}{15}x 41600 ]

[ x frac{41600 times 15}{52} Rightarrow x 12000 ]

Step 6: Calculating the Other Contributions

Now we substitute ( x 12000 ) back into the original equations to find the values of ( y ), ( C ), and ( z ):

( y frac{2}{3} times 12000 8000 )

( C 12000 3000 15000 )

( z frac{4}{5} times 15000 12000 )

Thus, the contributions are:

A: Sh. 12,000 B: Sh. 8,000 C: Sh. 15,000 D: Sh. 12,000

By following the steps and solving the equations, we can determine that the contributions are Sh. 12,000 for A, B, and D, and Sh. 15,000 for C. This scenario demonstrates the importance of understanding ratios and equations in business partnerships to ensure fair distribution of financial responsibilities.

Conclusion and Key Lessons

This mathematical problem-solving exercise not only helps in understanding business collaboration but also reinforces the importance of precise calculations in financial management. For aspiring entrepreneurs or anyone involved in business, mastering these mathematical skills can significantly enhance decision-making processes and financial planning capabilities. By applying these principles, one can ensure that business partnerships are structured fairly and efficiently.