Understanding Profit Sharing in Business Partnerships

When individuals start a business together, they often contribute different amounts of capital. The profit earned from the business is typically shared among the partners in proportion to the amount of money they invested and the duration for which the investment was made. In this specific problem, we are told that two individuals, A and B, start a business by investing different amounts. We need to determine how a total profit is divided between them based on their initial investments.

Calculating Investment Ratio

The key to solving this profit-sharing problem is to find the ratio of the investments made by A and B. The investments are:

• Investment by A = Rs. 1,00,000
• Investment by B = Rs. 1,50,000

The ratio of their investments (A : B) is:

$$ \text{Ratio} = \text{Investment of A} : \text{Investment of B} $$

$$ \text{Ratio} = 1,00,000 : 1,50,000 $$

To simplify this ratio, we can divide both numbers by their greatest common divisor. We can see that both numbers have five zeros, so we can divide by 1,00,000:

$$ \text{Ratio} = \frac{1,00,000}{10,000} : \frac{1,50,000}{10,000} $$

$$ \text{Ratio} = 10 : 15 $$

Now, we can further simplify the ratio 10 : 15 by dividing both numbers by 5:

$$ \text{Ratio} = \frac{10}{5} : \frac{15}{5} $$

$$ \text{Ratio} = 2 : 3 $$

So, the investments made by A and B are in the ratio 2 : 3.

Sharing the Total Profit Based on Ratio

The total profit earned by the business is Rs. 24,000. This total profit needs to be divided between A and B in the ratio of their investments, which is 2 : 3.

The sum of the parts in the ratio is $2 + 3 = 5$. This means the total profit is divided into 5 equal parts, where A gets 2 parts and B gets 3 parts.

Calculating A's Share of Profit

A's share of the profit is the fraction of their ratio part (2) over the total ratio parts (5), multiplied by the total profit.

$$ \text{A's Share} = \left( \frac{\text{A's Ratio Part}}{\text{Total Ratio Parts}} \right) \times \text{Total Profit} $$

$$ \text{A's Share} = \left( \frac{2}{5} \right) \times 24,000 $$

To calculate this, we can divide 24,000 by 5 and then multiply by 2:

$$ \frac{24,000}{5} = 4,800 $$

$$ \text{A's Share} = 2 \times 4,800 = 9,600 $$

So, A's share of the profit is Rs. 9,600.

Calculating B's Share of Profit

B's share of the profit is the fraction of their ratio part (3) over the total ratio parts (5), multiplied by the total profit.

$$ \text{B's Share} = \left( \frac{\text{B's Ratio Part}}{\text{Total Ratio Parts}} \right) \times \text{Total Profit} $$

$$ \text{B's Share} = \left( \frac{3}{5} \right) \times 24,000 $$

To calculate this, we can divide 24,000 by 5 and then multiply by 3:

$$ \frac{24,000}{5} = 4,800 $$

$$ \text{B's Share} = 3 \times 4,800 = 14,400 $$

So, B's share of the profit is Rs. 14,400.

The respective shares of A and B out of the total profit of Rs. 24,000 are Rs. 9,600 and Rs. 14,400.

Summary of Profit Shares

We can check if the sum of the shares equals the total profit: $9,600 + 14,400 = 24,000$. This matches the total profit given in the problem.

Revision Table: Key Concepts in Profit Sharing

Additional Information on Partnership Profits

In a simple partnership where investments are made at the start and remain constant, and all partners are involved for the same duration, the profit is typically divided strictly according to the ratio of their initial investments. This is because the investment is the basis for generating profit.

• If investments are for different durations, the profit sharing ratio is calculated based on the product of the investment amount and the time period for which it was invested (Investment $\times$ Time).
• Partnership agreements can sometimes specify a different method for profit sharing, such as including salaries for active partners or distributing a fixed percentage before the remaining profit is shared by investment ratio. However, for problems like this, the standard assumption is profit sharing based purely on the investment ratio when time is not mentioned or is the same for all.
• Understanding ratios is fundamental to solving problems related to partnerships and profit distribution. A ratio expresses the relative size of two or more values.

This problem illustrates a basic case of profit sharing in a business partnership, where the distribution is directly proportional to the capital contributed by each partner.