$A$ started a business with a capital of $\text{Rs. }1,12,000$. After $2$ months, $B$ joined the business with a capital of $\text{Rs. }80,000$, and after another $2$ months, $C$ joined the business with a capital of $\text{Rs. }72,000$. After $10$ months from the start of the business, $B$ withdraws $\text{Rs. }8,000$ and $C$ also withdraws $\text{Rs. }8,000$. If $B$ received $\text{Rs. }9,800$ as his share in the profit at the end of a year, then the total profit was:

To find the profit-sharing ratio, we calculate the equivalent capital for $1$ month (Investment $\times$ Duration in months) for each partner over a total period of $12$ months:
*For $A$ **: Invested $\text{Rs. }1,12,000$ for the entire $12$ months.

$${\text{Equivalent Capital}}_A=1,12,000\times 12=1,344,000$$

*For $B$ **: Joined after $2$ months (active for $10$ months total). Invested $\text{Rs. }80,000$ for $8$ months (from month $2$ to month $10$), then withdrew $\text{Rs. }8,000$, leaving $\text{Rs. }72,000$ for the remaining $2$ months.

$${\text{Equivalent Capital}}_B=(80,000\times 8)+(72,000\times 2)=640,000+144,000=784,000$$

*For $C$ **: Joined $2$ months after $B$ (active for $8$ months total). Invested $\text{Rs. }72,000$ for $6$ months (from month $4$ to month $10$), then withdrew $\text{Rs. }8,000$, leaving $\text{Rs. }64,000$ for the remaining $2$ months.

$${\text{Equivalent Capital}}_C=(72,000\times 6)+(64,000\times 2)=432,000+128,000=560,000$$

Now, find the ratio of their investments:

$$\text{Ratio }A:B:C=1,344,000:784,000:560,000=1344:784:560$$

Dividing by $112$:

$$\text{Ratio }A:B:C=12:7:5$$

Given that $B$ 's share of profit is $\text{Rs. }9,800$, which corresponds to $7$ units:

$$7\text{ units}=9,800⟹1\text{ unit}=\frac{9,800}{7}=1,400$$

The total profit corresponds to $(12+7+5)=24$ units:

$$\text{Total Profit}=24\times 1,400=\text{Rs. }33,600$$