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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.

Solve the following expression: (1/2 + 2/5) / (1/3 - 3)

Learn how to solve the expression (1/2 + 2/5) / (1/3 - 3) with step-by-step instructions. This solution involves adding and subtracting fractions, converting whole numbers to fractions, and dividing fractions. Perfect for students in Grades 6-8.

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