To simplify the given expression:

$\frac{\frac{7}{3} - \frac{3}{4}}{\frac{1}{5} + \frac{3}{4}}$

we will simplify both the numerator and the denominator separately before dividing them.

Step 1: Simplify the Numerator

The numerator is:

$\frac{7}{3} - \frac{3}{4}$

To subtract these fractions, we need a common denominator, which is 12 (the least common multiple of 3 and 4).

$\frac{7}{3} = \frac{7 \times 4}{3 \times 4} = \frac{28}{12}$

$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

Now, subtract the fractions:

$\frac{28}{12} - \frac{9}{12} = \frac{19}{12}$

Step 2: Simplify the Denominator

The denominator is:

$\frac{1}{5} + \frac{3}{4}$

We again need a common denominator, which is 20 (the least common multiple of 5 and 4).

$\frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20}$

$\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$

Now, add the fractions:

$\frac{4}{20} + \frac{15}{20} = \frac{19}{20}$

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now, divide the simplified numerator by the simplified denominator:

$\frac{\frac{19}{12}}{\frac{19}{20}} = \frac{19}{12} \times \frac{20}{19}$

Simplify by canceling out the 19s:

$\frac{20}{12} = \frac{5}{3}$

Thus, the simplified expression is:

$\frac{5}{3}$

Would you like further details, or do you have any questions?

Here are some related questions you might find interesting:

1. How do you find the least common multiple of two numbers?
2. How can you simplify complex fractions?
3. What are the steps for dividing fractions?
4. How do you handle mixed numbers in fraction operations?
5. What are some real-world applications of fraction simplification?

Tip: When working with fractions, always look for common denominators to simplify addition or subtraction!