Partnership Profit Calculation: Finding Investment Duration

This problem involves a partnership where partners A and B invest money for different periods, and the profit is shared based on the product of their investment amount and the duration of their investment. We are given the initial investment ratio and the final profit ratio, and we need to find out for how many months partner A invested.

Understanding Partnership Profit Sharing

In a partnership, the profit is generally distributed among the partners in proportion to the product of their respective investments and the time period for which the investment was made. This can be represented as:

Profit Ratio of A & B = (Investment of A × Time period of A) : (Investment of B × Time period of B)

Setting up the Investment Details

• The ratio of initial investments of A and B is given as 3 ∶ 5. Let the investment amount of A be \(3x\) and the investment amount of B be \(5x\), where \(x\) is a common factor.
• The partnership lasted for a year, which is 12 months.
• Partner B invested for the entire year. So, the time period for B's investment is 12 months.
• Partner A withdrew after a few months. Let the time period for A's investment be \(m\) months. We need to find the value of \(m\).

Calculating Capital-Time Products

The capital-time product for each partner is the amount invested multiplied by the duration of investment in months.

• Capital-Time Product for A = Investment of A × Time of A = \(3x \times m = 3xm\)
• Capital-Time Product for B = Investment of B × Time of B = \(5x \times 12 = 60x\)

Using the Profit Ratio to Find A's Investment Time

We are given that the profit was received in the ratio 2 ∶ 5 at the end of the year. According to the partnership principle, this profit ratio must be equal to the ratio of their capital-time products.

Profit Ratio of A & B = (Capital-Time Product of A) : (Capital-Time Product of B)

\(\frac{\text{Profit of A}}{\text{Profit of B}} = \frac{3xm}{60x}\)

We are given the Profit Ratio is 2 ∶ 5, so:

\(\frac{2}{5} = \frac{3xm}{60x}\)

Solving for A's Investment Duration (m)

Now, we need to solve the equation for \(m\). We can cancel the common factor \(x\) from the numerator and denominator on the right side:

\(\frac{2}{5} = \frac{3m}{60}\)

To solve for \(m\), we can cross-multiply:

\(2 \times 60 = 5 \times 3m\)

\(120 = 15m\)

Now, divide both sides by 15 to find \(m\):

\(m = \frac{120}{15}\)

\(m = 8\)

So, A invested for 8 months.

Final Answer Derivation

The calculation shows that A's investment duration \(m\) is 8 months. This means A invested their share for 8 months before withdrawing.

Revision Table: Key Partnership Concepts

Additional Information: Partnership Structures and Rules

Partnerships can have different structures and rules depending on the agreement:

• Simple Partnership: Partners invest for the same duration. Profit is shared in the ratio of their investments.
• Compound Partnership: Partners invest for different durations or change their investments during the period. Profit is shared in the ratio of their capital-time products. This problem is an example of a compound partnership.
• Active Partner: A partner who contributes capital and actively participates in the management of the business.
• Sleeping Partner: A partner who contributes capital but does not participate in the management.

The agreement between partners (the Partnership Deed) is crucial as it specifies the terms of investment, profit/loss sharing, withdrawal rules, and other operational details. If there is no specific agreement, rules based on relevant laws (like the Indian Partnership Act, 1932 in India) are applied, which often dictate equal profit sharing regardless of capital or time invested, unless explicitly agreed otherwise.