A, B, and C started a business with investments of Rs. 12,000, Rs. 12,000, and Rs. 8,000 respectively. B invested for only $x$ months, while C left the business $x$ months before the completion of one year. If A received Rs. 1800 out of the total annual profit of Rs. 3200, find the value of $x$.

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**Solution:**

1. Let the time period for B's investment be $x$ months.
2. Since C left $x$ months before the year ended, C invested for $12 - x$ months.
3. A invested Rs. 12,000 for 12 months, so A's investment-time product = $12,000 \times 12 = 144,000$.
4. B invested Rs. 12,000 for $x$ months, so B's investment-time product = $12,000 \times x = 12,000x$.
5. C invested Rs. 8,000 for $12 - x$ months, so C's investment-time product = $8,000 \times (12 - x) = 96,000 - 8,000x$.

6. The ratio of their profits will be proportional to their investment-time products:
   $$	ext{A : B : C} = 144,000 : 12,000x : (96,000 - 8,000x)$$

7. Given A's share of profit is Rs. 1800 out of total Rs. 3200, so:
   $$\frac{\text{A's share}}{\text{Total profit}} = \frac{1800}{3200} = \frac{9}{16}$$

8. The sum of the ratio parts is:
   $$144,000 + 12,000x + 96,000 - 8,000x = 240,000 + 4,000x$$

9. A's share ratio is:
   $$\frac{144,000}{240,000 + 4,000x} = \frac{9}{16}$$

10. Cross-multiplied:
    $$144,000 \times 16 = 9 \times (240,000 + 4,000x)$$
    $$2,304,000 = 2,160,000 + 36,000x$$

11. Simplify:
    $$2,304,000 - 2,160,000 = 36,000x$$
    $$144,000 = 36,000x$$

12. Solve for $x$:
    $$x = \frac{144,000}{36,000} = 4$$

**Answer:** $4$

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