Three partners X, Y and Z started their business by investing ₹40,000, ₹38,000 and ₹30,000, respectively. After 6 months, X and Z made additional investments of ₹20,000 and ₹15,000 respectively, whereas Y withdrew ₹8,000. Find the share of Y (in ₹) in the total profit of ₹38,880 made at the end of the year.
10,880
Partnership Profit Sharing Calculation
This problem involves calculating the profit share of partners in a business where their investments changed during the year. The profit is shared in proportion to the effective capital invested over the period of the business.
The total profit made at the end of the year is ₹38,880.
We need to calculate the equivalent investment for each partner for the entire year by considering the duration for which each amount was invested. The year has 12 months, and the investments changed after 6 months.
Calculating Effective Investment for Each Partner
Let's calculate the effective investment for X, Y, and Z over the 12-month period.
Partner X:
- Initial investment: ₹40,000 for the first 6 months.
- Additional investment after 6 months: ₹20,000.
- Investment for the next 6 months: ₹40,000 + ₹20,000 = ₹60,000.
- Effective investment for X = (Investment in Period 1 × Duration) + (Investment in Period 2 × Duration)
- Effective investment for X = (\(₹40,000 \times 6\)) + (\(₹60,000 \times 6\))
- Effective investment for X = \(₹240,000 + ₹360,000 = ₹600,000\).
Partner Y:
- Initial investment: ₹38,000 for the first 6 months.
- Withdrawal after 6 months: ₹8,000.
- Investment for the next 6 months: ₹38,000 - ₹8,000 = ₹30,000.
- Effective investment for Y = (Investment in Period 1 × Duration) + (Investment in Period 2 × Duration)
- Effective investment for Y = (\(₹38,000 \times 6\)) + (\(₹30,000 \times 6\))
- Effective investment for Y = \(₹228,000 + ₹180,000 = ₹408,000\).
Partner Z:
- Initial investment: ₹30,000 for the first 6 months.
- Additional investment after 6 months: ₹15,000.
- Investment for the next 6 months: ₹30,000 + ₹15,000 = ₹45,000.
- Effective investment for Z = (Investment in Period 1 × Duration) + (Investment in Period 2 × Duration)
- Effective investment for Z = (\(₹30,000 \times 6\)) + (\(₹45,000 \times 6\))
- Effective investment for Z = \(₹180,000 + ₹270,000 = ₹450,000\).
Determining the Profit Sharing Ratio
The profit sharing ratio among partners X, Y, and Z is the ratio of their effective investments:
Ratio X : Y : Z = Effective Investment of X : Effective Investment of Y : Effective Investment of Z
Ratio X : Y : Z = \(₹600,000 : ₹408,000 : ₹450,000\)
We can simplify this ratio by dividing all numbers by common factors.
Divide by 1000: \(600 : 408 : 450\)
Divide by 6: \(100 : 68 : 75\)
The simplified profit sharing ratio is \(100 : 68 : 75\).
Total ratio parts = \(100 + 68 + 75 = 243\).
Calculating Y's Share of Profit
Y's share of the total profit is given by:
Y's Share = \(\left(\frac{\text{Y's ratio part}}{\text{Total ratio parts}}\right) \times \text{Total Profit}\)
Y's Share = \(\left(\frac{68}{243}\right) \times ₹38,880\)
Calculate the value:
\(\frac{38,880}{243} = 160\)
Y's Share = \(68 \times 160 = ₹10,880\)
Therefore, the share of Y in the total profit is ₹10,880.
| Partner | Investment (0-6 months) | Investment (6-12 months) | Effective Annual Investment (Investment × Time) |
|---|---|---|---|
| X | ₹40,000 | ₹60,000 | (₹40,000 × 6) + (₹60,000 × 6) = ₹600,000 |
| Y | ₹38,000 | ₹30,000 | (₹38,000 × 6) + (₹30,000 × 6) = ₹408,000 |
| Z | ₹30,000 | ₹45,000 | (₹30,000 × 6) + (₹45,000 × 6) = ₹450,000 |
Revision Table: Partnership Profit Calculation
| Concept | Description | Application in Problem |
|---|---|---|
| Partnership | Business run by two or more individuals agreeing to share profits or losses. | X, Y, and Z are partners in a business. |
| Profit Sharing Ratio | The ratio in which partners share the profit or loss. Usually based on the ratio of (Investment × Time). | Calculated as Effective Investment Ratio X:Y:Z = 100:68:75. |
| Effective Investment | Total capital contributed considering the duration of investment. For varying investments, it's the sum of (Investment amount × time duration). | Calculated for each partner over the 12 months. |
Additional Information: Partnership Concepts
Partnership problems often involve calculating profit or loss shares based on the partners' investments and the time period for which the investment was made. If investments are constant for the entire duration, the profit is shared simply in the ratio of investments. However, if investments change or partners join/leave mid-way, the effective investment (capital multiplied by the time it was invested) is used to determine the ratio.
- Simple Partnership: Investment is constant for the same duration for all partners. Profit is shared in the ratio of their investments.
- Compound Partnership: Investments vary over time or partners invest for different durations. Profit is shared in the ratio of the product of their investment amounts and the respective time periods.
- In this problem, since investments changed after 6 months, it is an example of a compound partnership. The effective investment for each partner is calculated as the sum of (investment amount × time duration) for each distinct period of investment.
Important Questions from Partnership
A, B and C invested their capitals in the ratio 2 ∶ 3 ∶ 5. The ratio of months for which they invested is 4 ∶ 2 ∶ 3, respectively. If the difference between the profit shares of A and B is Rs. 1,86,000, then C's share of profit (in Rs.) is:
A started a business with a capital of Rs. 54,000 and admitted B and C after 4 months and 6 months, respectively. At the end of the year, the profit was divided among the three in the ratio 1 ∶ 4 ∶ 5. What is the sum (in Rs.) of the capitals invested by B and C?
A, B and C started a business in partnership. Initially, A invested Rs. 29,000, while B and C invested Rs. 25,000 each. After 4 months, A withdrew Rs. 3,000. After 2 more months, C invested Rs. 12,000 more. Find the share of C( in Rs.) in the profit of Rs. 33,200 at the end of the year.
A, B and C invest in a business in the ratio 4 ∶ 5 ∶ 7. C is a sleeping partner, so his share of profits will be half of what it would have been if he were a working partner. If they make Rs 36,000 profit of which 25% is reinvested in the business, how much does B get (in Rs)?
A and B entered into a partnership with investments in the ratio 3 ∶ 5. After a few months, A withdrew and collected his money back. At the end of the year, they received profit in the ratio 2 ∶ 5. For how many months did A invest?
