Students can practice their arithmetic skills by solving these 15 fraction division problems to find the correct quotients.

To divide fractions, you follow a simple rule: Keep, Change, Flip.

1. Keep the first fraction exactly as it is.
2. Change the division sign ($\div$) to a multiplication sign ($\times$).
3. Flip the second fraction (swap the top and bottom numbers). This is called finding the reciprocal.

Then, multiply the top numbers together and the bottom numbers together. Finally, simplify the answer if possible.

Here are the step-by-step solutions for each problem:

1. $\frac{5}{9} \div \frac{2}{5}$
* Keep $\frac{5}{9}$, Change to $\times$, Flip $\frac{2}{5}$ to $\frac{5}{2}$.
* $\frac{5}{9} \times \frac{5}{2} = \frac{5 \times 5}{9 \times 2} = \frac{25}{18}$
* As a mixed number: $1 \frac{7}{18}$

2. $\frac{6}{8} \div \frac{2}{3}$
* First, simplify $\frac{6}{8}$ to $\frac{3}{4}$ to make it easier.
* Keep $\frac{3}{4}$, Change to $\times$, Flip $\frac{2}{3}$ to $\frac{3}{2}$.
* $\frac{3}{4} \times \frac{3}{2} = \frac{3 \times 3}{4 \times 2} = \frac{9}{8}$
* As a mixed number: $1 \frac{1}{8}$

3. $\frac{2}{3} \div \frac{3}{6}$
* First, simplify $\frac{3}{6}$ to $\frac{1}{2}$.
* Keep $\frac{2}{3}$, Change to $\times$, Flip $\frac{1}{2}$ to $\frac{2}{1}$.
* $\frac{2}{3} \times \frac{2}{1} = \frac{4}{3}$
* As a mixed number: $1 \frac{1}{3}$

4. $\frac{5}{6} \div \frac{3}{8}$
* Keep $\frac{5}{6}$, Change to $\times$, Flip $\frac{3}{8}$ to $\frac{8}{3}$.
* $\frac{5}{6} \times \frac{8}{3} = \frac{40}{18}$
* Simplify by dividing top and bottom by 2: $\frac{20}{9}$
* As a mixed number: $2 \frac{2}{9}$

5. $\frac{2}{3} \div \frac{3}{8}$
* Keep $\frac{2}{3}$, Change to $\times$, Flip $\frac{3}{8}$ to $\frac{8}{3}$.
* $\frac{2}{3} \times \frac{8}{3} = \frac{16}{9}$
* As a mixed number: $1 \frac{7}{9}$

6. $\frac{8}{9} \div \frac{1}{3}$
* Keep $\frac{8}{9}$, Change to $\times$, Flip $\frac{1}{3}$ to $\frac{3}{1}$.
* $\frac{8}{9} \times \frac{3}{1} = \frac{24}{9}$
* Simplify by dividing top and bottom by 3: $\frac{8}{3}$
* As a mixed number: $2 \frac{2}{3}$

7. $\frac{1}{2} \div \frac{2}{4}$
* Notice that $\frac{2}{4}$ is the same as $\frac{1}{2}$. Any number divided by itself is 1.
* Let's calculate anyway: Keep $\frac{1}{2}$, Change to $\times$, Flip $\frac{2}{4}$ to $\frac{4}{2}$.
* $\frac{1}{2} \times \frac{4}{2} = \frac{4}{4} = 1$

8. $\frac{2}{4} \div \frac{3}{9}$
* Simplify first: $\frac{2}{4} = \frac{1}{2}$ and $\frac{3}{9} = \frac{1}{3}$.
* Keep $\frac{1}{2}$, Change to $\times$, Flip $\frac{1}{3}$ to $\frac{3}{1}$.
* $\frac{1}{2} \times \frac{3}{1} = \frac{3}{2}$
* As a mixed number: $1 \frac{1}{2}$

9. $\frac{1}{3} \div \frac{3}{4}$
* Keep $\frac{1}{3}$, Change to $\times$, Flip $\frac{3}{4}$ to $\frac{4}{3}$.
* $\frac{1}{3} \times \frac{4}{3} = \frac{4}{9}$

10. $\frac{6}{8} \div \frac{2}{8}$
* Since the denominators (bottoms) are the same, you can just divide the tops: $6 \div 2 = 3$.
* Or use the rule: Keep $\frac{6}{8}$, Change to $\times$, Flip $\frac{2}{8}$ to $\frac{8}{2}$.
* $\frac{6}{8} \times \frac{8}{2} = \frac{48}{16} = 3$

11. $\frac{6}{8} \div \frac{1}{6}$
* Simplify $\frac{6}{8}$ to $\frac{3}{4}$.
* Keep $\frac{3}{4}$, Change to $\times$, Flip $\frac{1}{6}$ to $\frac{6}{1}$.
* $\frac{3}{4} \times \frac{6}{1} = \frac{18}{4}$
* Simplify by dividing by 2: $\frac{9}{2}$
* As a mixed number: $4 \frac{1}{2}$

12. $\frac{2}{5} \div \frac{4}{6}$
* Simplify $\frac{4}{6}$ to $\frac{2}{3}$.
* Keep $\frac{2}{5}$, Change to $\times$, Flip $\frac{2}{3}$ to $\frac{3}{2}$.
* $\frac{2}{5} \times \frac{3}{2} = \frac{6}{10}$
* Simplify by dividing by 2: $\frac{3}{5}$

13. $\frac{4}{5} \div \frac{2}{5}$
* Denominators are the same, so just divide tops: $4 \div 2 = 2$.
* Or use rule: $\frac{4}{5} \times \frac{5}{2} = \frac{20}{10} = 2$

14. $\frac{1}{2} \div \frac{1}{4}$
* Keep $\frac{1}{2}$, Change to $\times$, Flip $\frac{1}{4}$ to $\frac{4}{1}$.
* $\frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2$

15. $\frac{1}{2} \div \frac{7}{9}$
* Keep $\frac{1}{2}$, Change to $\times$, Flip $\frac{7}{9}$ to $\frac{9}{7}$.
* $\frac{1}{2} \times \frac{9}{7} = \frac{9}{14}$

Final Answer:
1. $\frac{25}{18}$ or $1 \frac{7}{18}$
2. $\frac{9}{8}$ or $1 \frac{1}{8}$
3. $\frac{4}{3}$ or $1 \frac{1}{3}$
4. $\frac{20}{9}$ or $2 \frac{2}{9}$
5. $\frac{16}{9}$ or $1 \frac{7}{9}$
6. $\frac{8}{3}$ or $2 \frac{2}{3}$
7. $1$
8. $\frac{3}{2}$ or $1 \frac{1}{2}$
9. $\frac{4}{9}$
10. $3$
11. $\frac{9}{2}$ or $4 \frac{1}{2}$
12. $\frac{3}{5}$
13. $2$
14. $2$
15. $\frac{9}{14}$