To determine how many cups of sugar are needed for 8 cookies, we can set up a proportion based on the fact that 1 1/4 cups of sugar are required for 20 cookies.

First, convert 1 1/4 to an improper fraction: $1 \frac{1}{4} = \frac{5}{4}$

Now, set up a proportion: $\frac{5}{4} \text{ cups of sugar for 20 cookies} = \frac{x}{8} \text{ cups of sugar for 8 cookies}$

Solve for $x$ by cross-multiplying: $5 \times 8 = 4 \times 20x$

Simplify the equation: $40 = 160x$

Now divide both sides by 160 to isolate $x$: $x = \frac{40}{160} = \frac{1}{4}$

Thus, you need 1/4 cup of sugar to make 8 cookies.

Would you like more details on how this was calculated or have any questions?

Here are 5 questions for further exploration:

1. How would the amount of sugar change if you wanted to make 12 cookies?
2. What if you wanted to make 40 cookies?
3. How many cups of sugar would you need to make just 1 cookie?
4. Can you express the relationship between cookies and sugar as a unit rate?
5. How many cups of sugar are needed for 100 cookies?

Tip: Converting mixed numbers to improper fractions makes it easier to solve ratio problems like these!