Let's solve the given problem step-by-step.

The problem is:

$\frac{\frac{1}{2} + \frac{2}{5}}{\frac{1}{3} - 3}$

First, we need to simplify the numerator and the denominator separately.

Simplifying the Numerator:

$\frac{1}{2} + \frac{2}{5}$

To add these fractions, we need a common denominator. The least common multiple (LCM) of 2 and 5 is 10.

Convert each fraction to have a denominator of 10: $\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$ $\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$

Now add the fractions: $\frac{5}{10} + \frac{4}{10} = \frac{9}{10}$

Simplifying the Denominator:

$\frac{1}{3} - 3$

Convert 3 to a fraction with the same denominator: $3 = \frac{3 \times 3}{1 \times 3} = \frac{9}{3}$

Now subtract the fractions: $\frac{1}{3} - \frac{9}{3} = \frac{1 - 9}{3} = \frac{-8}{3}$

Putting it all together:

$\frac{\frac{9}{10}}{\frac{-8}{3}}$

To divide by a fraction, multiply by its reciprocal: $\frac{9}{10} \times \frac{3}{-8} = \frac{9 \times 3}{10 \times -8} = \frac{27}{-80} = -\frac{27}{80}$

So the simplified form of the expression is: $-\frac{27}{80}$

Would you like any details or have any questions about this solution?

Here are five relative questions to further your understanding:

1. How do you find the least common multiple (LCM) of two numbers?
2. What is the process for converting a whole number to a fraction?
3. How can you simplify a complex fraction?
4. What are some common mistakes to avoid when adding or subtracting fractions?
5. How do you multiply and divide fractions?

Tip: Always simplify fractions at each step to make calculations easier and avoid mistakes.