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P, Q and R enter into a partnership with...

P, Q and R enter into a partnership with capitals in the ratio `3 : 2 : 1`. After 4 months, P leves the business and after 4 more months Q also leaves the business and R continues till the end of the year. If R takes `10%` of the profit for managing the business, then what part of the profit does R get?

A

`37%`

B

`36%`

C

`27%`

D

`30%`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will first determine the effective capital contribution of each partner over time, then calculate the total profit, and finally find out how much profit R receives after taking into account the management fee. ### Step 1: Determine the capital contributions The capitals of P, Q, and R are in the ratio 3:2:1. Let's assume the capitals are: - P's capital = 3x - Q's capital = 2x - R's capital = 1x ### Step 2: Calculate the time each partner's capital was invested - P invests for 4 months (then leaves). - Q invests for 8 months (4 months after P leaves). - R invests for the entire year (12 months). ### Step 3: Calculate the effective capital contribution Now we calculate the effective capital contribution for each partner: - P's contribution = 3x * 4 months = 12x - Q's contribution = 2x * 8 months = 16x - R's contribution = 1x * 12 months = 12x ### Step 4: Calculate the total capital contribution Total capital contribution = P's contribution + Q's contribution + R's contribution = 12x + 16x + 12x = 40x ### Step 5: Calculate the share of each partner in the profit Now we find the ratio of their contributions: - P's share = 12x / 40x = 3/10 - Q's share = 16x / 40x = 2/5 - R's share = 12x / 40x = 3/10 ### Step 6: Calculate R's profit after management fee Let’s assume the total profit is P. R takes 10% of the profit for managing the business: - R's management fee = 10% of P = 0.1P The remaining profit after deducting R's management fee: - Remaining profit = P - 0.1P = 0.9P Now, R's share of the remaining profit: - R's share of remaining profit = (3/10) * 0.9P = 0.27P ### Step 7: Calculate R's total profit Finally, R's total profit = R's management fee + R's share of remaining profit: - R's total profit = 0.1P + 0.27P = 0.37P ### Conclusion R gets 37% of the total profit.
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