Let's evaluate the given expression step-by-step:

$\frac{3}{4} \div \frac{2}{3} + \frac{1}{2} \div \frac{3}{4}$

Step 1: Divide $\frac{3}{4}$ by $\frac{2}{3}$

When dividing fractions, we multiply by the reciprocal of the second fraction:

$\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{3 \times 3}{4 \times 2} = \frac{9}{8}$

Step 2: Divide $\frac{1}{2}$ by $\frac{3}{4}$

Again, multiply by the reciprocal of the second fraction:

$\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6}$

Simplify $\frac{4}{6}$ by dividing both the numerator and denominator by 2:

$\frac{4}{6} = \frac{2}{3}$

Step 3: Add $\frac{9}{8}$ and $\frac{2}{3}$

To add these fractions, find a common denominator. The least common denominator (LCD) of 8 and 3 is 24.

Convert both fractions to have a denominator of 24: $\frac{9}{8} = \frac{27}{24}, \quad \frac{2}{3} = \frac{16}{24}$

Now add them:

$\frac{27}{24} + \frac{16}{24} = \frac{43}{24}$

Final Answer:

The result of the expression is:

$\frac{43}{24}$

This is an improper fraction. You can convert it to a mixed number:

$\frac{43}{24} = 1 \frac{19}{24}$

Answer: $1 \frac{19}{24}$

Would you like a more detailed explanation or have further questions?

Here are 5 related questions you might find interesting:

1. How do you multiply two fractions?
2. What is the procedure for simplifying a fraction?
3. What is the least common denominator (LCD), and how do you find it?
4. How do you convert an improper fraction to a mixed number?
5. What happens if you multiply instead of divide fractions?

Tip: To avoid errors in fraction division, always remember to multiply by the reciprocal of the second fraction.