Printable math worksheet for practicing addition of two proper fractions.

Problem Overview:

The task involves adding two proper fractions and simplifying the results where necessary. Proper fractions are fractions where the numerator is less than the denominator.

Solution Approach:

1. Addition of Fractions:
- If the denominators are the same, simply add the numerators and keep the denominator.
- If the denominators are different, find a common denominator, convert each fraction to have this common denominator, add the numerators, and then simplify if possible.

2. Simplification:
- After adding, check if the resulting fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Step-by-Step Solutions:

#### Section 1: Problems 1–10

1. Problem 1: $\frac{3}{6} + \frac{1}{6}$
- Denominators are the same.
- Add the numerators: $3 + 1 = 4$.
- Result: $\frac{4}{6}$.
- Simplify: $\frac{4}{6} = \frac{2}{3}$ (divide numerator and denominator by 2).
- Answer: $\frac{4}{6} = \frac{2}{3}$.

2. Problem 2: $\frac{6}{9} + \frac{1}{5}$
- Denominators are different. Find the least common denominator (LCD): $45$.
- Convert fractions:
- $\frac{6}{9} = \frac{6 \times 5}{9 \times 5} = \frac{30}{45}$,
- $\frac{1}{5} = \frac{1 \times 9}{5 \times 9} = \frac{9}{45}$.
- Add the fractions: $\frac{30}{45} + \frac{9}{45} = \frac{39}{45}$.
- Simplify: $\frac{39}{45} = \frac{13}{15}$ (divide numerator and denominator by 3).
- Answer: $\frac{39}{45} = \frac{13}{15}$.

3. Problem 3: $\frac{1}{3} + \frac{1}{3}$
- Denominators are the same.
- Add the numerators: $1 + 1 = 2$.
- Result: $\frac{2}{3}$.
- No simplification needed.
- Answer: $\frac{2}{3}$.

4. Problem 4: $\frac{2}{5} + \frac{1}{5}$
- Denominators are the same.
- Add the numerators: $2 + 1 = 3$.
- Result: $\frac{3}{5}$.
- No simplification needed.
- Answer: $\frac{3}{5}$.

5. Problem 5: $\frac{6}{8} + \frac{1}{8}$
- Denominators are the same.
- Add the numerators: $6 + 1 = 7$.
- Result: $\frac{7}{8}$.
- No simplification needed.
- Answer: $\frac{7}{8}$.

6. Problem 6: $\frac{1}{5} + \frac{1}{5}$
- Denominators are the same.
- Add the numerators: $1 + 1 = 2$.
- Result: $\frac{2}{5}$.
- No simplification needed.
- Answer: $\frac{2}{5}$.

7. Problem 7: $\frac{7}{9} + \frac{1}{9}$
- Denominators are the same.
- Add the numerators: $7 + 1 = 8$.
- Result: $\frac{8}{9}$.
- No simplification needed.
- Answer: $\frac{8}{9}$.

8. Problem 8: $\frac{2}{7} + \frac{3}{7}$
- Denominators are the same.
- Add the numerators: $2 + 3 = 5$.
- Result: $\frac{5}{7}$.
- No simplification needed.
- Answer: $\frac{5}{7}$.

9. Problem 9: $\frac{2}{6} + \frac{3}{6}$
- Denominators are the same.
- Add the numerators: $2 + 3 = 5$.
- Result: $\frac{5}{6}$.
- No simplification needed.
- Answer: $\frac{5}{6}$.

10. Problem 10: $\frac{2}{6} + \frac{2}{6}$
- Denominators are the same.
- Add the numerators: $2 + 2 = 4$.
- Result: $\frac{4}{6}$.
- Simplify: $\frac{4}{6} = \frac{2}{3}$ (divide numerator and denominator by 2).
- Answer: $\frac{4}{6} = \frac{2}{3}$.

#### Section 2: Problems 11–20

11. Problem 11: $\frac{4}{6} + \frac{1}{6}$
- Denominators are the same.
- Add the numerators: $4 + 1 = 5$.
- Result: $\frac{5}{6}$.
- No simplification needed.
- Answer: $\frac{5}{6}$.

12. Problem 12: $\frac{4}{8} + \frac{2}{8}$
- Denominators are the same.
- Add the numerators: $4 + 2 = 6$.
- Result: $\frac{6}{8}$.
- Simplify: $\frac{6}{8} = \frac{3}{4}$ (divide numerator and denominator by 2).
- Answer: $\frac{6}{8} = \frac{3}{4}$.

13. Problem 13: $\frac{3}{9} + \frac{3}{9}$
- Denominators are the same.
- Add the numerators: $3 + 3 = 6$.
- Result: $\frac{6}{9}$.
- Simplify: $\frac{6}{9} = \frac{2}{3}$ (divide numerator and denominator by 3).
- Answer: $\frac{6}{9} = \frac{2}{3}$.

14. Problem 14: $\frac{2}{8} + \frac{4}{8}$
- Denominators are the same.
- Add the numerators: $2 + 4 = 6$.
- Result: $\frac{6}{8}$.
- Simplify: $\frac{6}{8} = \frac{3}{4}$ (divide numerator and denominator by 2).
- Answer: $\frac{6}{8} = \frac{3}{4}$.

15. Problem 15: $\frac{2}{4} + \frac{1}{4}$
- Denominators are the same.
- Add the numerators: $2 + 1 = 3$.
- Result: $\frac{3}{4}$.
- No simplification needed.
- Answer: $\frac{3}{4}$.

16. Problem 16: $\frac{3}{7} + \frac{1}{7}$
- Denominators are the same.
- Add the numerators: $3 + 1 = 4$.
- Result: $\frac{4}{7}$.
- No simplification needed.
- Answer: $\frac{4}{7}$.

17. Problem 17: $\frac{2}{5} + \frac{1}{5}$
- Denominators are the same.
- Add the numerators: $2 + 1 = 3$.
- Result: $\frac{3}{5}$.
- No simplification needed.
- Answer: $\frac{3}{5}$.

18. Problem 18: $\frac{2}{7} + \frac{1}{7}$
- Denominators are the same.
- Add the numerators: $2 + 1 = 3$.
- Result: $\frac{3}{7}$.
- No simplification needed.
- Answer: $\frac{3}{7}$.

19. Problem 19: $\frac{2}{8} + \frac{5}{8}$
- Denominators are the same.
- Add the numerators: $2 + 5 = 7$.
- Result: $\frac{7}{8}$.
- No simplification needed.
- Answer: $\frac{7}{8}$.

20. Problem 20: $\frac{1}{4} + \frac{1}{4}$
- Denominators are the same.
- Add the numerators: $1 + 1 = 2$.
- Result: $\frac{2}{4}$.
- Simplify: $\frac{2}{4} = \frac{1}{2}$ (divide numerator and denominator by 2).
- Answer: $\frac{2}{4} = \frac{1}{2}$.

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & \frac{4}{6} = \frac{2}{3} \\
2. & \frac{39}{45} = \frac{13}{15} \\
3. & \frac{2}{3} \\
4. & \frac{3}{5} \\
5. & \frac{7}{8} \\
6. & \frac{2}{5} \\
7. & \frac{8}{9} \\
8. & \frac{5}{7} \\
9. & \frac{5}{6} \\
10. & \frac{4}{6} = \frac{2}{3} \\
11. & \frac{5}{6} \\
12. & \frac{6}{8} = \frac{3}{4} \\
13. & \frac{6}{9} = \frac{2}{3} \\
14. & \frac{6}{8} = \frac{3}{4} \\
15. & \frac{3}{4} \\
16. & \frac{4}{7} \\
17. & \frac{3}{5} \\
18. & \frac{3}{7} \\
19. & \frac{7}{8} \\
20. & \frac{2}{4} = \frac{1}{2} \\
\end{array}
}
\]