Andy had 1600 white marbles and some black marbles at first. After buying 1200 red marbles, $\frac{5}{9}$ of his marbles were black marbles and red marbles. What fraction of the marbles were red in the end?

1. Identify the fraction of white marbles in the end

Since $\frac{5}{9}$ of the marbles in the end were black and red, the remaining fraction must be white marbles.
Fraction of white marbles = $1 - \frac{5}{9} = \frac{4}{9}$

2. Find the total number of marbles in the end

The number of white marbles remained unchanged at 1600.
So, $\frac{4}{9}$ of the total marbles = 1600.
$\frac{1}{9}$ of the total marbles = $1600 \div 4 = 400$.
Total marbles in the end = $400 \times 9 = 3600$.

3. Calculate the fraction of red marbles in the end

Number of red marbles = 1200.
Fraction of red marbles = $\frac{1200}{3600} = \frac{1}{3}$.