Vijay and Manohar share profits and losses in the ratio of $2:1$. They admit Prakash as a partner with a $\frac{1}{4}$ share in profits, guaranteeing that his annual share of profit will be at least $\text{Rs. }50,000$. The net profit of the firm for the financial year ending March 31, 2015, was $\text{Rs. }1,60,000$. Calculate the final amount of profit Vijay will receive.

Let us calculate the profit distribution step by step:Calculate Prakash's Share from the Net Profit:
$$\text{Prakash’s share}=1,60,000\times \frac{1}{4}=\text{Rs. }40,000$$
Evaluate the Guaranteed Amount and Shortfall:Prakash was guaranteed a minimum of $\text{Rs. }50,000$.
$$\text{Deficiency}=\text{Guaranteed Minimum}-\text{Actual Share}=50,000-40,000=\text{Rs. }10,000$$
Since the problem does not specify a separate ratio for covering deficiencies, the remaining partners (Vijay and Manohar) will bear this $\text{Rs. }10,000$ shortfall in their old relative ratio ($2:1$).Calculate the Remaining Profit to be Split Between Vijay and Manohar:
$$\text{Remaining Profit}=\text{Total Profit}-\text{Prakash’s Actual Share}=1,60,000-40,000=\text{Rs. }1,20,000$$
Calculate Vijay's Share Before the Deficiency Adjustment:
$$\text{Vijay’s Initial Share}=1,20,000\times \frac{2}{3}=\text{Rs. }80,000$$
Deduct Vijay's Contribution to Prakash's Shortfall:
$$\text{Vijay’s Contribution}=10,000\times \frac{2}{3}=\text{Rs. }6,666.67\approx \text{Rs. }6,667$$
$$\text{Vijay’s Final Profit}=80,000-6,666.67=\text{Rs. }73,333.33\approx \text{Rs. }73,333$$