A and B entered into a business with investments of Rs $P$ and Rs $P + 5000$ respectively. After four months, A withdrew 40\% of his initial investment, and C joined the business with an investment of Rs 6000. At the end of one year, the total profit is Rs 17150, and C received $\frac{60}{343}$th of the total profit. Find the profit share of B (in Rs).

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Accepted Answer

**Solution:**

1. **Given:**
   - A's initial investment = Rs $P$
   - B's investment = Rs $P + 5000$
   - After 4 months, A withdraws 40\% of his initial investment.
   - C joins after 4 months with Rs 6000.
   - Total profit at the end of 1 year = Rs 17150
   - C's share of profit = $\frac{60}{343}$ of total profit

2. **Calculate C's profit:**
   $$
   \text{C's profit} = \frac{60}{343} \times 17150 = \frac{60 \times 17150}{343}
   $$

Calculate numerator:
   $$60 \times 17150 = 1,029,000$$

Divide by 343:
   $$\frac{1,029,000}{343} = 3000$$

So, C's profit = Rs 3000

3. **Calculate C's investment-time product:**
   - C invested Rs 6000 for 8 months (since C joined after 4 months)
   - C's capital-time product = $6000 \times 8 = 48000$

4. **Calculate A's investment-time product:**
   - For first 4 months, A invested Rs $P$
   - After 4 months, A withdrew 40\% of $P$, so new investment = 60\% of $P$ = $0.6P$
   - For next 8 months, investment = $0.6P$

So, A's capital-time product:
   $$
   (P \times 4) + (0.6P \times 8) = 4P + 4.8P = 8.8P
   $$

5. **Calculate B's investment-time product:**
   - B invested Rs $P + 5000$ for 12 months
   - So, B's capital-time product = $(P + 5000) \times 12 = 12P + 60000$

6. **Total capital-time product:**
   $$
   8.8P + 12P + 60000 + 48000 = (8.8P + 12P) + (60000 + 48000) = 20.8P + 108000
   $$

7. **Profit sharing ratio:**
   Profit is divided in the ratio of capital-time products.

C's share ratio:
   $$
   \frac{C's \ capital-time}{Total \ capital-time} = \frac{48000}{20.8P + 108000}
   $$

Given C's profit share fraction is $\frac{60}{343}$, so:
   $$
   \frac{48000}{20.8P + 108000} = \frac{60}{343}
   $$

8. **Solve for $P$:**
   Cross multiply:
   $$
   48000 \times 343 = 60 \times (20.8P + 108000)
   $$

Calculate left side:
   $$48000 \times 343 = 16,464,000$$

So,
   $$
   16,464,000 = 60 \times 20.8P + 60 \times 108000
   $$
   $$
   16,464,000 = 1248P + 6,480,000
   $$

Subtract 6,480,000 from both sides:
   $$
   16,464,000 - 6,480,000 = 1248P
   $$
   $$
   9,984,000 = 1248P
   $$

Divide both sides by 1248:
   $$
   P = \frac{9,984,000}{1248} = 8000
   $$

9. **Calculate B's capital-time product:**
   $$
   12P + 60000 = 12 \times 8000 + 60000 = 96000 + 60000 = 156000
   $$

10. **Calculate total capital-time product:**
    $$
    20.8P + 108000 = 20.8 \times 8000 + 108000 = 166400 + 108000 = 274400
    $$

11. **Calculate B's profit share:**
    $$
    \text{B's share fraction} = \frac{156000}{274400} = \frac{156}{274.4} = \frac{390}{686} = \frac{195}{343}
    $$

B's profit = $\frac{195}{343} \times 17150$

Calculate numerator:
    $$195 \times 17150 = 3,344,250$$

Divide by 343:
    $$\frac{3,344,250}{343} = 9750$$

**Answer:** Rs $9750$

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