Let's go through the questions step by step.

Question 6:

$\frac{2}{5} \div \frac{2}{3} = \frac{2}{5} \times \frac{3}{2} = \frac{2 \times 3}{5 \times 2} = \frac{6}{10} = \frac{3}{5}$ Answer: $\frac{3}{5}$

Question 7:

$\frac{1}{5} + 4 \div \frac{2}{3} = \frac{1}{5} + \frac{4}{1} \times \frac{3}{2} = \frac{1}{5} + \frac{12}{2} = \frac{1}{5} + 6$ Converting 6 into a fraction: $\frac{1}{5} + \frac{30}{5} = \frac{31}{5}$ Answer: $\frac{31}{5}$

Question 8:

$2 + \frac{5}{4} = \frac{8}{4} + \frac{5}{4} = \frac{13}{4}$ The next part is: $\frac{13}{4} \times 4 = \frac{13}{4} \times \frac{4}{1} = \frac{13 \times 4}{4} = 13$ Answer: 13

Question 9:

$\frac{5}{9} + 2 \div \frac{2}{3} = \frac{5}{9} + 2 \times \frac{3}{2} = \frac{5}{9} + 3 = \frac{5}{9} + \frac{27}{9} = \frac{32}{9}$ Answer: $\frac{32}{9}$

Question 10:

$\frac{2}{7} \div \frac{4 \frac{2}{3}}{1} = \frac{2}{7} \times \frac{3}{14} = \frac{2 \times 3}{7 \times 14} = \frac{6}{98} = \frac{3}{49}$ Answer: $\frac{3}{49}$

Do you need further clarifications on any of the steps?

Here are some related questions you might find interesting:

1. How do you multiply fractions in different denominators?
2. What is the inverse operation for division involving fractions?
3. How can mixed numbers be converted into improper fractions?
4. What happens when you multiply two fractions with large numerators?
5. What are some tips to simplify complex fractions?

Tip: Always simplify fractions at each step to make calculations easier!