P, Q and R started a business each investing Rs. 20,000. After 5 months, P withdrew Rs. 5,000, Q withdrew Rs. 4,000 and R invested Rs. 6,000 more. At the end of the year, a total profit of Rs. 34,950 was recorded. Find the share of P.
Understanding Partnership Profit Sharing
In a business partnership, profits are typically shared among partners based on their investments and the duration for which the investments were made. This is often represented as the ratio of the product of investment amount and time period for each partner.
In this problem, three partners P, Q, and R started a business. Their investments changed after 5 months. The total duration of the business is 1 year, which is 12 months.
Calculating Each Partner's Effective Investment Time
We need to calculate the total 'investment-time' for each partner by summing up the product of the investment amount and the time period for which that amount was invested.
Initial investment by P, Q, R = Rs. 20,000 each.
This initial investment lasted for 5 months.
After 5 months (for the remaining $12 - 5 = 7$ months):
- P withdrew Rs. 5,000. P's investment for the next 7 months = Rs. $20,000 - 5,000 = \text{Rs. } 15,000$.
- Q withdrew Rs. 4,000. Q's investment for the next 7 months = Rs. $20,000 - 4,000 = \text{Rs. } 16,000$.
- R invested Rs. 6,000 more. R's investment for the next 7 months = Rs. $20,000 + 6,000 = \text{Rs. } 26,000$.
Now, let's calculate the total investment-time for each partner:
- P's total investment-time: $(20000 \times 5) + (15000 \times 7)$
$= 100000 + 105000 = 205000$ - Q's total investment-time: $(20000 \times 5) + (16000 \times 7)$
$= 100000 + 112000 = 212000$ - R's total investment-time: $(20000 \times 5) + (26000 \times 7)$
$= 100000 + 182000 = 282000$
Determining the Profit Sharing Ratio
The ratio of profits will be equal to the ratio of their total investment-time products.
Ratio of P : Q : R = $205000 : 212000 : 282000$
We can simplify this ratio by dividing each number by 1000:
Ratio of P : Q : R = $205 : 212 : 282$
Calculating P's Share of Total Profit
The total profit recorded at the end of the year is Rs. 34,950.
The sum of the ratios is $205 + 212 + 282 = 699$.
P's share of the profit is calculated as:
$\text{P's Share} = \left(\frac{\text{P's Ratio}}{\text{Sum of Ratios}}\right) \times \text{Total Profit}$
$\text{P's Share} = \left(\frac{205}{699}\right) \times 34950$
Let's calculate the value:
$\frac{34950}{699} = 50$
$\text{P's Share} = 205 \times 50 = 10250$
So, P's share of the total profit is Rs. 10,250.
Final Result
The share of P in the total profit of Rs. 34,950 is Rs. 10,250.
Revision Table: Partnership Calculations
| Partner | Initial Investment (Rs.) | Investment Duration 1 (Months) | Changed Investment (Rs.) | Investment Duration 2 (Months) | Total Investment-Time |
|---|---|---|---|---|---|
| P | 20,000 | 5 | 15,000 | 7 | $(20000 \times 5) + (15000 \times 7) = 205000$ |
| Q | 20,000 | 5 | 16,000 | 7 | $(20000 \times 5) + (16000 \times 7) = 212000$ |
| R | 20,000 | 5 | 26,000 | 7 | $(20000 \times 5) + (26000 \times 7) = 282000$ |
Additional Information on Partnership Profit Sharing
Profit sharing in a partnership is governed by the partnership deed. If there is no deed or the deed doesn't specify the profit-sharing ratio, profits are generally shared equally among partners, regardless of their capital contributions. However, when investments vary in amount or duration, profits are shared in the ratio of the equivalent capital invested for the same time period. This equivalent capital for a given time period is calculated as Investment Amount $\times$ Time Period.
Key points:
- Profits and losses are shared in the agreed ratio.
- If investments change, the calculation must account for the different amounts and the periods they were invested.
- The ratio of (Investment $\times$ Time) for each partner determines the profit-sharing ratio when investments vary over time.
Important Questions from Partnership
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.
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)?
