Let's break down the expression step by step:

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

Step 1: Simplify inside the brackets

Start with the parentheses inside the brackets: $2 + 3 = 5$ Now substitute back into the expression: $\frac{1}{2} \left[1 - 2 \times 5 \times 20 \right] / 2$

Step 2: Perform the multiplication inside the brackets

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

Step 3: Simplify inside the brackets

$1 - 200 = -199$ Now substitute back: $\frac{1}{2} \left[-199 \right] / 2$

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'd like any further explanation or details, feel free to ask!

Here are some related questions:

1. How do you simplify expressions involving fractions?
2. What is the order of operations in mathematical expressions?
3. How do you handle parentheses in complex algebraic expressions?
4. How do you multiply fractions with whole numbers?
5. What are the rules for dealing with negative numbers in algebra?
6. How do you divide fractions by whole numbers?
7. How do you convert a fraction to a decimal?
8. How can you check your work in algebraic simplifications?

Tip: Always remember to follow the order of operations (PEMDAS/BODMAS) to avoid mistakes in simplifications.