Making proportions equal

Question

Consider this problem:

A bag is filled with wheat and rice in proportions of 3 and 5 respectively, and they are completely mixed. What proportion of the bag should be emptied and replaced with wheat so that the proportion of wheat and rice become equal?

My attempt:

Let $x$ be the amount of wheat we add to the bag. So,

$$\frac{3}{8} + x - \frac{x}{2} = \frac{5}{8} - \frac{x}{2}$$

I subtracted $\frac{x}{2}$ because they are mixed and half of what we take out is wheat. It results $x=\frac{1}{4}$, but the correct answer is $\frac{1}{5}$ why?

Answer 1

The correct solution would be $$\frac{3}{8} + x - \frac{3x}{8} = \frac{5}{8} - \frac{5x}{8}$$ which leads to $x=\frac{1}{5}$.

Answer 2

You've assumed that we take out an equal quantity of rice and wheat when we remove some of the mixture; in fact, we take out more rice than we do wheat.