Understanding Profit Sharing Ratios

This problem involves dividing a total profit based on a given ratio between two individuals, A and B. A profit sharing ratio tells us how a total amount is to be split among partners. If the ratio is x : 2, it means that for every 'x' parts of the profit A receives, B receives 2 parts.

Analyzing the Given Profit Distribution Problem

We are given the following information:

• Total Profit = Rs. 240
• A's Share of Profit = Rs. 80
• Ratio of Profit Distribution between A and B = x : 2

Our goal is to find the value of x in the profit sharing ratio.

Setting up the Equation for Profit Ratio

The share of an individual in the profit is proportional to their part in the ratio, relative to the sum of the ratio parts. The total ratio parts in this case are x + 2.

A's share can be represented as the fraction of the total profit that corresponds to A's part in the ratio:

<b>A's Share / Total Profit = A's Ratio Part / (Sum of Ratio Parts)</b>

Substituting the given values and the ratio:

\(\frac{80}{240} = \frac{x}{x + 2}\)

Solving for the Unknown Value x

Now we need to solve the equation for x. First, simplify the fraction on the left side:

\(\frac{80}{240} = \frac{80 \div 80}{240 \div 80} = \frac{1}{3}\)

So the equation becomes:

\(\frac{1}{3} = \frac{x}{x + 2}\)

To solve for x, we can cross-multiply:

\(1 \times (x + 2) = 3 \times x\)

\(x + 2 = 3x\)

Now, rearrange the equation to isolate x terms on one side:

\(2 = 3x - x\)

\(2 = 2x\)

Divide both sides by 2:

\(x = \frac{2}{2}\)

\(x = 1\)

The value of x in the profit sharing ratio is 1.

Verifying the Solution

If x = 1, the profit sharing ratio is 1 : 2. The total ratio parts are 1 + 2 = 3.

A's share would be \(\frac{1}{3}\) of the total profit.

A's Share = \(\frac{1}{3} \times 240 = \frac{240}{3} = 80\)

This matches the given information that A got Rs. 80 as his share. The solution is correct.

Revision Table: Key Concepts

Additional Information on Profit Distribution

Profit distribution is a fundamental concept in partnerships and business. It ensures that profits are allocated fairly among contributors, often based on their investment, time contribution, or a pre-agreed ratio. Ratios provide a simple way to represent these proportions. When one value (like one partner's share or the total profit) is known along with the ratio, we can often find the other unknown values, including an unknown part within the ratio itself, as demonstrated in this problem.