Determining the Capital Investment in a Partnership

When partners come together to initiate a business venture, the distribution of profits and capital investments plays a crucial role in maintaining a fair and balanced partnership. This article provides a step-by-step guide to determining C's capital investment in a partnership where A and B have made specific investments, leading to a particular profit distribution.

The Problem Overview

Three partners, A, B, and C, enter into a partnership. A invests $2560, B invests $2000, and C invests an as-of-yet unknown amount. At the end of the year, the total profit is $1105, with A receiving $320. The challenge is to determine the amount invested by C.

Step-by-Step Solution

1. **Determine the Profit and Share:** - Total profit for the year $1105 - Share of A $320 2. **Find the Profit Share Ratio:** Using the ratio of A's investment to the total investment, we first calculate A's share of the total profit based on the given amounts. Let's denote C's investment as (x). - Total investment $2560 $2000 $x $4560 $x - A's share of the profit (frac{2560}{4560 x} times 1105 320) 3. **Cross-Multiply to Solve for (x):** Cross-multiplying the equation we get: [ 2560 times 1105 320 times (4560 x) ] Simplifying the right side: [ 2828800 1459200 32 ] Isolating (32): [ 2828800 - 1459200 32 ] [ 1369600 32 ] Dividing both sides by 320 to solve for (x): [ x frac{1369600}{320} 4280 ] Therefore, C's capital investment is $4280.

Alternative Method

Alternatively, we can use a simplified approach to solve for C's capital investment. Given the distribution and ratios, we can directly find the value of (x) that satisfies all conditions.

1. **First, calculate A's profit distribution ratio:** Given: [ frac{2560}{4560 x} times 1105 320 ] Simplifying: [ frac{2560}{4560 x} frac{320}{1105} ] Solve for (x) by cross-multiplying and isolating (x): - From the simplified ratio, we can set up the equation and solve for (x) as above.

2. **Using A and B's profit distribution:** Given: [ frac{2560}{4560 x} times 1105 320 ] Simplifying step-by-step to find (x 4280).

Conclusion

The solution confirms that C's capital investment in the partnership is $4280. This methodology ensures a clear understanding of how the profit distribution and investment ratios are interconnected and provides a sound basis for resolving such partnership scenarios with accuracy.

Key Takeaways

1. **Profit Distribution:** Properly understanding and calculating profit distribution based on investment ratios is crucial in maintaining a fair partnership environment. 2. **Investment Ratios:** The ratio of each partner's investment directly impacts their share of the profit, making accurate calculation essential. 3. **Mathematical Approach:** Using algebraic methods to solve for unknowns in partnership financial scenarios ensures a precise and reliable outcome.