Calculating New Profit Ratio for Partners

This problem involves calculating the new profit sharing ratio among partners after their initial investments are increased by different percentages. In a business partnership where the time period of investment is the same for all partners (as stated, "for one year"), the profit sharing ratio is directly proportional to the investment ratio. Therefore, we first need to find the new investment amounts after the increases and then determine the ratio of these new investments.

Understanding the Initial Investments and Increases

The initial investment ratio of Suresh, Dinesh, and Ramesh is given as $3 : 6 : 8$. Let the initial investments be:

• Suresh's initial investment: $3k$
• Dinesh's initial investment: $6k$
• Ramesh's initial investment: $8k$

where $k$ is a proportionality constant.

The investments are increased by the following percentages:

• Suresh's investment: Increased by 5%
• Dinesh's investment: Increased by 15%
• Ramesh's investment: Increased by 20%

Calculating New Investments

To find the new investment for each partner, we add the percentage increase to their initial investment.

• Suresh's new investment: Initial investment $\times$ (1 + Percentage increase) ${3k \times (1 + 5\%)} = {3k \times (1 + \frac{5}{100})} = {3k \times (1 + 0.05)} = {3k \times 1.05} = {3.15k}$
• Dinesh's new investment: Initial investment $\times$ (1 + Percentage increase) ${6k \times (1 + 15\%)} = {6k \times (1 + \frac{15}{100})} = {6k \times (1 + 0.15)} = {6k \times 1.15} = {6.90k}$
• Ramesh's new investment: Initial investment $\times$ (1 + Percentage increase) ${8k \times (1 + 20\%)} = {8k \times (1 + \frac{20}{100})} = {8k \times (1 + 0.20)} = {8k \times 1.20} = {9.60k}$

Determining the New Investment Ratio

The new investment ratio for Suresh, Dinesh, and Ramesh is the ratio of their new investments:

Suresh : Dinesh : Ramesh ${=} {3.15k : 6.90k : 9.60k}$

We can cancel out the proportionality constant $k$:

New Investment Ratio ${=} {3.15 : 6.90 : 9.60}$

To express this ratio in simplest whole numbers, we can multiply each part by a suitable number to remove the decimals. Multiplying by 100 would give:

${315 : 690 : 960}$

Now, we find the greatest common divisor (GCD) of 315, 690, and 960 to simplify the ratio.

• Factors of 315: 1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315
• Factors of 690: 1, 2, 3, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690
• Factors of 960: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480, 960

The common factors are 1, 3, 5, 15. The GCD is 15.

Divide each part of the ratio ${315 : 690 : 960}$ by 15:

• ${315 \div 15} = {21}$
• ${690 \div 15} = {46}$
• ${960 \div 15} = {64}$

The simplified new investment ratio is ${21 : 46 : 64}$.

Final Profit Ratio

Since the time period of investment is the same for all partners (one year), the ratio of their profits will be the same as the ratio of their investments.

Therefore, the ratio of their profits for one year will be ${21 : 46 : 64}$.

The ratio of new investments is $3.15 : 6.90 : 9.60$. Multiplying by 100 gives $315 : 690 : 960$. Dividing by their GCD, 15, yields the simplified ratio $21 : 46 : 64$. This is the new profit ratio.

Revision Table: Partnership Profit Sharing

Additional Information: Ratio and Proportion in Business

Ratios are fundamental in partnership accounting and business mathematics. They help in fairly distributing profits or losses, managing investments, and understanding the relative contributions of partners. When investments change, the profit sharing ratio needs to be adjusted accordingly to reflect the new capital structure, assuming other factors like partner salaries or interest on capital are not specified as affecting the distribution method. Understanding how percentage changes affect ratios is crucial for solving problems like this.

A ratio represents a comparison between two or more quantities of the same kind. A proportion is an equality of two ratios. In partnership problems, ratios are used to represent the relationship between partners' investments or profits. Adjusting investments by percentages means calculating new values and then finding the ratio of these new values. This new ratio then determines how profits will be shared.