Simplification of Continued Fractions

In this problem, we're asked to evaluate a complex fraction, also known as a nested or continued fraction. The best way to solve these is to start from the bottom and work our way up.

Let's write down the expression clearly. We have one plus, one over, two plus, one over, three plus one fourth.

We start at the very bottom with three plus one fourth. To combine these, we convert three into the fraction twelve over four.

Twelve over four plus one over four gives us thirteen quarters.

Now, we have one divided by thirteen quarters. Dividing by a fraction is the same as multiplying by its reciprocal, so it becomes four over thirteen.

Now we simplify the denominator of the main fraction: two plus four over thirteen.