It looks like there's a minor issue with the expression. Let me clarify the steps and solve it for you.

Given expression:
$\frac{1}{2} \left[1 - 2(2 + 3) \times 20 \right) / 2$

This expression seems to have an extra closing parenthesis. If I assume the expression should be:

$\frac{1}{2} \times \left[1 - 2(2 + 3) \times 20 \right] / 2$

I'll proceed with this interpretation.

Step 1: Simplify inside the parentheses

$2 + 3 = 5$ So the expression becomes: $\frac{1}{2} \times \left[1 - 2 \times 5 \times 20 \right] / 2$

Step 2: Perform the multiplication inside the brackets

$2 \times 5 = 10$ Then: $10 \times 20 = 200$ So the expression now is: $\frac{1}{2} \times \left[1 - 200 \right] / 2$

Step 3: Simplify inside the brackets

$1 - 200 = -199$ Now the expression is: $\frac{1}{2} \times (-199) / 2$

Step 4: Multiply by $\frac{1}{2}$

$\frac{-199}{2} / 2$

Step 5: Divide by 2

$\frac{-199}{4}$

So, the final simplified expression is: $\boxed{-\frac{199}{4}}$

If you need any further clarification or if there's any other way you'd like to look at the problem, let me know!

Here are some related questions:

1. How do you handle expressions with nested parentheses?
2. What is the significance of the order of operations?
3. How do you simplify complex fractions?
4. How do you approach algebraic expressions with mixed operations?
5. What is the process for multiplying fractions with integers?
6. How do you simplify negative fractions?
7. How can you verify your result in algebraic simplifications?
8. What are some common mistakes to avoid in algebraic simplifications?

Tip: Always double-check your parentheses in complex expressions to avoid mistakes.