Thank you for pointing that out. Let's re-evaluate the given expression step by step correctly.

The given expression is:

$\frac{1}{2 - \frac{1}{1 + \frac{1}{2}}}$

First, let's simplify the innermost fraction:

$1 + \frac{1}{2} = 1 + 0.5 = 1.5 = \frac{3}{2}$

Now substitute this back into the expression:

$\frac{1}{2 - \frac{1}{\frac{3}{2}}}$

Next, simplify the denominator:

$\frac{1}{\frac{3}{2}} = \frac{2}{3}$

So, the expression becomes:

$\frac{1}{2 - \frac{2}{3}}$

Now, subtract $\frac{2}{3}$ from 2:

$2 = \frac{6}{3}$

$2 - \frac{2}{3} = \frac{6}{3} - \frac{2}{3} = \frac{4}{3}$

So, the expression now is:

$\frac{1}{\frac{4}{3}}$

Finally, simplify the fraction:

$\frac{1}{\frac{4}{3}} = \frac{3}{4}$

Thus, the simplified, reduced rational number is:

$\boxed{\frac{3}{4}}$

Would you like more details or have any questions?

Here are 5 more questions you might be interested in:

1. How do you solve quadratic equations by factoring?
2. What is the process to find the derivative of a function?
3. How do you simplify complex fractions?
4. What are the steps to solve systems of equations using the substitution method?
5. How do you convert a repeating decimal into a fraction?

Tip: When simplifying complex fractions, always work from the innermost fraction outward, simplifying step by step.