Let, Aarushi and Anushka have the same amount of money in their pocket, which is Rs.$$x$$.
After Anushka gives $$\frac13^{rd}$$ of the money to Aarushi-
Amount with Aarushi= $$\left ( x+\cfrac{x}{3} \right )$$
As per question-Aarushi gave a party at a restaurant and cleared the bill by paying half of the total money with her. Remaining money in Aarushi's pocket is Rs.$$1600$$
$$\Rightarrow \left ( x+\cfrac{x}{3} \right )-\cfrac{1}{2}\times\left ( x+\cfrac{x}{3} \right )=1600$$
[Taking $$\left(x+\cfrac{x}{3}\right)$$ common]
$$\Rightarrow \left(x+\cfrac{x}{3}\right)\left(1-\cfrac{1}{2}\right)=1600$$
$$\Rightarrow \left(\cfrac{3x+x}{3}\right)\left(\cfrac{2-1}{2}\right)=1600$$
$$\Rightarrow \cfrac{4x}{3}\times\cfrac{1}{2}=1600$$
$$\Rightarrow \cfrac{4x}{6}=1600$$
$$\Rightarrow 4x=9600$$
$$\Rightarrow x=\cfrac{9600}{4}=2400$$
Therefore, the equal amount of money in Anushka and Aarushi's pocket is Rs.$$2400$$.
Money gifted by Anushka-
$$\Rightarrow \cfrac{1}{3}$$ of $$2400$$
$$\Rightarrow \cfrac{1}{3}\times2400$$
$$\Rightarrow 800$$
Therefore, Anushka gave Rs.$$800$$ to Aarushi.