The expression in the image is:

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

Let's solve this step by step.

Step 1: Simplify the denominator

We need to simplify the denominator first:

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

This can be written as:

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

So, now the expression becomes:

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

Step 2: Simplify the fraction

To simplify $\frac{1}{\frac{3}{2}}$, we take the reciprocal of $\frac{3}{2}$, which is $\frac{2}{3}$. So, the expression becomes:

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

Step 3: Perform the subtraction

Now, subtract $\frac{2}{3}$ from 1. First, express 1 as a fraction with the same denominator:

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

So, we have:

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

Final Answer:

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

Would you like more details or further clarifications?

Here are five related questions to explore further:

1. How do you handle more complex nested fractions?
2. How do you subtract fractions with different denominators?
3. What are the general steps for simplifying compound fractions?
4. Can you convert mixed numbers into improper fractions to simplify expressions like this?
5. How would the result change if we used different values in the denominator?

Tip: Always simplify the inner fractions first when dealing with complex or nested fractions to make the calculation easier.