A, B and C started a business with investment of Rs. 12,000, Rs. 12,000 and Rs. 8,000 respectively. B invested only for ‘x’ months, while C left the business ‘x’ month before a year. If A got Rs 1800 out of annual profit of Rs 3200, then find the value of ‘x’?

The correct option is 4.

Let us solve the problem step-by-step by determining the ratio of profits shared among A, B, and C based on their investment amounts and investment durations.

Step 1: Write down the investments and periods for each person.
The business runs for a total of 1 year (12 months).
- Investment of A, $I_A=12000$ for $T_A=12$ months.
- Investment of B, $I_B=12000$ for $T_B=x$ months.
- Investment of C, $I_C=8000$. Since C left the business x months before the year ended, C invested for $T_C=12-x$ months.

Step 2: Calculate the product of investment and time for each person.
- For A: $12000\times 12=144000$
- For B: $12000\times x=12000x$
- For C: $8000\times (12-x)=96000-8000x$

Step 3: Determine total equivalent investment.
Total equivalent units = $144000+12000x+(96000-8000x)$

$\text{Total equivalent units}=240000+4000x$

Step 4: Use A's profit share to form an equation.
We are given:
- Total profit = Rs. 3200
- A's share of profit = Rs. 1800
Therefore, the ratio of A's investment-time product to the total investment-time product is equal to the ratio of A's profit to total profit:

$\frac{144000}{240000+4000x}=\frac{1800}{3200}$

Simplify the right hand side:
$\frac{1800}{3200}=\frac{9}{16}$

Simplify the left hand side by dividing the numerator and denominator by 4000:
$\frac{36}{60+x}=\frac{9}{16}$

Step 5: Solve for x.
Divide both sides of the equation by 9:
$\frac{4}{60+x}=\frac{1}{16}$

Cross-multiplying gives:
$60+x=4\times 16$

$60+x=64$

$x=64-60$

$x=4$

Thus, the value of x is 4.