P and Q invested Rs. 40000 and Rs. 75000 respectively in a business. After $(x + 2)$ months, P left the business, and simultaneously R joined by investing Rs. 50000. At the end of 1 year from the start, Q received Rs. 54000 out of the total profit of Rs. 87000. Find the value of $(x + 7)$.

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**Solution:**

1. Let the total time be 12 months.

2. P invested Rs. 40000 for $(x + 2)$ months.

3. Q invested Rs. 75000 for the entire 12 months.

4. R invested Rs. 50000 for the remaining $(12 - (x + 2)) = (10 - x)$ months.

5. The profit sharing ratio is proportional to the product of investment and time period.

- P's share = $40000 \times (x + 2)$

- Q's share = $75000 \times 12 = 900000$

- R's share = $50000 \times (10 - x)$

6. Total profit = Rs. 87000, and Q's share of profit = Rs. 54000.

7. The ratio of Q's share to total profit is:

$$\frac{Q's\ share}{Total\ profit} = \frac{75000 \times 12}{40000(x+2) + 75000 \times 12 + 50000(10 - x)} = \frac{54000}{87000} = \frac{54}{87} = \frac{18}{29}$$

8. Set up the equation:

$$\frac{900000}{40000(x+2) + 900000 + 50000(10 - x)} = \frac{18}{29}$$

9. Let the denominator be $D = 40000(x+2) + 900000 + 50000(10 - x)$

Simplify $D$:

$$D = 40000x + 80000 + 900000 + 500000 - 50000x = (40000x - 50000x) + (80000 + 900000 + 500000) = -10000x + 1,480,000$$

10. Substitute back:

$$\frac{900000}{-10000x + 1,480,000} = \frac{18}{29}$$

11. Cross multiply:

$$900000 \times 29 = 18 \times (-10000x + 1,480,000)$$

$$26,100,000 = -180,000x + 26,640,000$$

12. Rearranging:

$$-180,000x = 26,100,000 - 26,640,000 = -540,000$$

$$x = \frac{-540,000}{-180,000} = 3$$

13. Finally, find $x + 7 = 3 + 7 = 10$.

**Answer:** 10

**Note:** The correct option is (e) 10. However, since the options provided have 10 as option (e), but the marked correct answer in options is 14, there is a mismatch. The calculation shows $x + 7 = 10$, so the correct answer is option (e) 10.

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