P and Q started a business together. P's investment is Rs. 2000 more than Q's investment. After 8 months, P left the business. At the end of 1 year, the total profit earned is Rs. 6800, and P's share of the profit is Rs. 400X. Given that $6X^3 = 3090 - \sqrt{324}$, find the investment made by Q.

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Accepted Answer

**Solution:**

1. Given that P's investment is Rs. 2000 more than Q's investment. Let Q's investment be *x*.
   Then, P's investment = *x + 2000*.

2. P invests for 8 months, Q invests for 12 months.

3. Profit sharing is proportional to the product of investment and time.

4. P's share of profit = Rs. 400X, total profit = Rs. 6800.

5. Calculate $X$ from the given equation:
   $$6X^3 = 3090 - \sqrt{324}$$
   Since $\sqrt{324} = 18$,
   $$6X^3 = 3090 - 18 = 3072$$
   $$X^3 = \frac{3072}{6} = 512$$
   $$X = \sqrt[3]{512} = 8$$

6. Therefore, P's profit share = $400 	imes 8 = 3200$.

7. Q's profit share = $6800 - 3200 = 3600$.

8. Using the ratio of profit shares:
   $$\frac{P's\ share}{Q's\ share} = \frac{P's\ investment 	imes 8}{Q's\ investment 	imes 12}$$
   $$\frac{3200}{3600} = \frac{(x + 2000) 	imes 8}{x 	imes 12}$$

9. Simplify the ratio:
   $$\frac{8(x + 2000)}{12x} = \frac{3200}{3600} = \frac{8}{9}$$

10. Cross multiply:
    $$8(x + 2000) 	imes 9 = 12x 	imes 8$$
    $$72(x + 2000) = 96x$$

11. Expand and simplify:
    $$72x + 144000 = 96x$$
    $$144000 = 96x - 72x = 24x$$
    $$x = \frac{144000}{24} = 6000$$

12. But this contradicts the options. Check step 10 carefully:

Actually, step 9 ratio simplification was incorrect. Let's re-express step 9:

$$\frac{3200}{3600} = \frac{8}{9}$$ (correct)

So,
    $$\frac{8(x + 2000)}{12x} = \frac{8}{9}$$

Multiply both sides by $12x$:
    $$8(x + 2000) = \frac{8}{9} 	imes 12x$$

Simplify right side:
    $$8(x + 2000) = \frac{8}{9} 	imes 12x = \frac{96x}{9} = \frac{32x}{3}$$

13. Multiply both sides by 3:
    $$3 	imes 8(x + 2000) = 32x$$
    $$24(x + 2000) = 32x$$

14. Expand:
    $$24x + 48000 = 32x$$

15. Rearrange:
    $$48000 = 32x - 24x = 8x$$

16. Solve for *x*:
    $$x = \frac{48000}{8} = 6000$$

17. So, Q's investment is Rs. 6000.

**Answer:** Rs. 6000

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