Let's break down the problem step by step:

Step 1: Determine the total amount of flour Mrs. Phua bought.

She bought 2 bags of flour, each weighing $2 \frac{1}{2}$ kg.

$2 \times 2 \frac{1}{2} = 2 \times \frac{5}{2} = 5 \text{ kg}$

So, the total amount of flour she bought is 5 kg.

Step 2: Calculate the amount of flour left after baking the cakes.

She used $1 \frac{5}{6}$ kg to bake some cakes.

$\text{Flour left} = 5 - 1 \frac{5}{6} = 5 - \frac{11}{6} = \frac{30}{6} - \frac{11}{6} = \frac{19}{6} \text{ kg}$

Step 3: Calculate the amount of flour used to bake the cookies.

She used $\frac{3}{4}$ of the remaining flour to bake cookies.

$\text{Flour used for cookies} = \frac{3}{4} \times \frac{19}{6} = \frac{3 \times 19}{4 \times 6} = \frac{57}{24} = \frac{19}{8} \text{ kg}$

Step 4: Simplify the fraction.

$\frac{19}{8} \text{ kg} = 2 \frac{3}{8} \text{ kg}$

So, Mrs. Phua used $2 \frac{3}{8}$ kg of flour to bake the cookies.

Would you like more details on this, or do you have any questions?

Here are some related questions you might find interesting:

1. How much flour did Mrs. Phua have left after baking both the cakes and cookies?
2. If she baked another batch of cookies using half of the remaining flour, how much flour would be left?
3. What percentage of the total flour was used to bake the cakes?
4. How much flour would she need if she wanted to bake another cake using the same recipe?
5. If the weight of the bags were different, how would that affect the amount of flour used for cookies?

Tip: When working with fractions, it's often helpful to convert mixed numbers to improper fractions to simplify calculations.