Question

Two friends P and Q started a business investing in the ratio of 5 : 6. R joined them after six months investing an amount equal to that of Q's. At the end of the year, 20% of profit was earned, which was equal to Rs. 98000. Determine the amount invested by R ?

• A. Rs. 105000
• B. Rs. 420000
• C. Rs. 200000
• D. Rs. 110000
• E. Rs. 210000

Hint:

In partnership problems, the profit-sharing ratio is based on the product of investment and time. If the profit is given, work backwards to find the actual investment values.

Answer 1

Answer: (E)

Step 1: Understand the Profit Distribution:
Profit is shared in the ratio of (Investment $\times$ Tim(e).
Step 2: Let the Investments:
Let P's investment = $5x$, Q's investment = $6x$. R invested an amount equal to Q's = $6x$. R joined after 6 months, so R's time = $12 - 6 = 6$ months. P and Q invested for the full year = 12 months.
Step 3: Calculate Profit-Sharing Ratio:
P's share ratio = $5x \times 12 = 60x$. Q's share ratio = $6x \times 12 = 72x$. R's share ratio = $6x \times 6 = 36x$. Ratio = $60x : 72x : 36x = 60 : 72 : 36 = 5 : 6 : 3$ (dividing by 12).
Step 4: Find Total Profit:
20% of profit = Rs. 98000. So, Total Profit = $\frac{98000}{0.2} = 98000 \times 5 = \text{Rs. } 4,90,000$.
Step 5: Find Value of One Part:
Total parts in ratio = $5+6+3 = 14$. One part = $\frac{4,90,000}{14} = 35,000$.
Step 6: Find R's Investment:
R's share of profit = $3 \times 35,000 = \text{Rs. } 1,05,000$. But this is R's profit, not investment. The ratio is for profit, not for investment value directly. We need to find the actual amount invested by R. We have the ratio of profit shares (5:6:3). Let the actual profit be in the same ratio. So R's profit = $105000$. But R's profit = (R's investment) $\times$ (R's time factor). R's time factor is 6 months = 0.5 years. Let R's investment = $I_R$. Then profit share is proportional to $I_R \times 0.5$. Similarly, Q's profit share = $6 \times 35,000 = 210,000 = (6x) \times 1$. Thus, $6x = 210,000 \implies x = 35,000$. Then R's investment = $6x = 6 \times 35,000 = 210,000$.
Step 7: Final Answer:
The amount invested by R is Rs. 2,10,000.