Adding Fractions Worksheet 3 - Solve the equations with fraction addition problems.

To solve the given problems involving the addition of fractions, we need to follow these steps:

1. Find a Common Denominator: If the denominators are different, find a common denominator (preferably the least common denominator).
2. Adjust the Fractions: Rewrite each fraction with the common denominator.
3. Add the Numerators: Add the numerators while keeping the denominator the same.
4. Simplify the Result: Simplify the resulting fraction if possible.

Let's solve each problem step by step.

---

Problem 1: $\frac{1}{3} + \frac{1}{9}$

- Step 1: The denominators are 3 and 9. The least common denominator (LCD) is 9.
- Step 2: Rewrite $\frac{1}{3}$ with a denominator of 9:
$$
\frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}
$$
- Step 3: Add the fractions:
$$
\frac{3}{9} + \frac{1}{9} = \frac{3 + 1}{9} = \frac{4}{9}
$$
- Step 4: The fraction $\frac{4}{9}$ is already in simplest form.
- Answer: $\boxed{\frac{4}{9}}$

---

Problem 2: $\frac{1}{5} + \frac{5}{10}$

- Step 1: The denominators are 5 and 10. The LCD is 10.
- Step 2: Rewrite $\frac{1}{5}$ with a denominator of 10:
$$
\frac{1}{5} = \frac{1 \times 2}{5 \times 2} = \frac{2}{10}
$$
- Step 3: Add the fractions:
$$
\frac{2}{10} + \frac{5}{10} = \frac{2 + 5}{10} = \frac{7}{10}
$$
- Step 4: The fraction $\frac{7}{10}$ is already in simplest form.
- Answer: $\boxed{\frac{7}{10}}$

---

Problem 3: $\frac{1}{6} + \frac{1}{18}$

- Step 1: The denominators are 6 and 18. The LCD is 18.
- Step 2: Rewrite $\frac{1}{6}$ with a denominator of 18:
$$
\frac{1}{6} = \frac{1 \times 3}{6 \times 3} = \frac{3}{18}
$$
- Step 3: Add the fractions:
$$
\frac{3}{18} + \frac{1}{18} = \frac{3 + 1}{18} = \frac{4}{18}
$$
- Step 4: Simplify $\frac{4}{18}$:
$$
\frac{4}{18} = \frac{2}{9}
$$
- Answer: $\boxed{\frac{2}{9}}$

---

Problem 4: $\frac{1}{2} + \frac{2}{6}$

- Step 1: The denominators are 2 and 6. The LCD is 6.
- Step 2: Rewrite $\frac{1}{2}$ with a denominator of 6:
$$
\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}
$$
- Step 3: Add the fractions:
$$
\frac{3}{6} + \frac{2}{6} = \frac{3 + 2}{6} = \frac{5}{6}
$$
- Step 4: The fraction $\frac{5}{6}$ is already in simplest form.
- Answer: $\boxed{\frac{5}{6}}$

---

Problem 5: $\frac{1}{3} + \frac{3}{12}$

- Step 1: The denominators are 3 and 12. The LCD is 12.
- Step 2: Rewrite $\frac{1}{3}$ with a denominator of 12:
$$
\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}
$$
- Step 3: Add the fractions:
$$
\frac{4}{12} + \frac{3}{12} = \frac{4 + 3}{12} = \frac{7}{12}
$$
- Step 4: The fraction $\frac{7}{12}$ is already in simplest form.
- Answer: $\boxed{\frac{7}{12}}$

---

Problem 6: $\frac{1}{4} + \frac{3}{8}$

- Step 1: The denominators are 4 and 8. The LCD is 8.
- Step 2: Rewrite $\frac{1}{4}$ with a denominator of 8:
$$
\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}
$$
- Step 3: Add the fractions:
$$
\frac{2}{8} + \frac{3}{8} = \frac{2 + 3}{8} = \frac{5}{8}
$$
- Step 4: The fraction $\frac{5}{8}$ is already in simplest form.
- Answer: $\boxed{\frac{5}{8}}$

---

Problem 7: $\frac{3}{8} + \frac{4}{24}$

- Step 1: The denominators are 8 and 24. The LCD is 24.
- Step 2: Rewrite $\frac{3}{8}$ with a denominator of 24:
$$
\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}
$$
- Step 3: Add the fractions:
$$
\frac{9}{24} + \frac{4}{24} = \frac{9 + 4}{24} = \frac{13}{24}
$$
- Step 4: The fraction $\frac{13}{24}$ is already in simplest form.
- Answer: $\boxed{\frac{13}{24}}$

---

Problem 8: $\frac{1}{4} + \frac{2}{12}$

- Step 1: The denominators are 4 and 12. The LCD is 12.
- Step 2: Rewrite $\frac{1}{4}$ with a denominator of 12:
$$
\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}
$$
- Step 3: Add the fractions:
$$
\frac{3}{12} + \frac{2}{12} = \frac{3 + 2}{12} = \frac{5}{12}
$$
- Step 4: The fraction $\frac{5}{12}$ is already in simplest form.
- Answer: $\boxed{\frac{5}{12}}$

---

Problem 9: $\frac{1}{7} + \frac{1}{21}$

- Step 1: The denominators are 7 and 21. The LCD is 21.
- Step 2: Rewrite $\frac{1}{7}$ with a denominator of 21:
$$
\frac{1}{7} = \frac{1 \times 3}{7 \times 3} = \frac{3}{21}
$$
- Step 3: Add the fractions:
$$
\frac{3}{21} + \frac{1}{21} = \frac{3 + 1}{21} = \frac{4}{21}
$$
- Step 4: The fraction $\frac{4}{21}$ is already in simplest form.
- Answer: $\boxed{\frac{4}{21}}$

---

Problem 10: $\frac{1}{5} + \frac{2}{20}$

- Step 1: The denominators are 5 and 20. The LCD is 20.
- Step 2: Rewrite $\frac{1}{5}$ with a denominator of 20:
$$
\frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20}
$$
- Step 3: Add the fractions:
$$
\frac{4}{20} + \frac{2}{20} = \frac{4 + 2}{20} = \frac{6}{20}
$$
- Step 4: Simplify $\frac{6}{20}$:
$$
\frac{6}{20} = \frac{3}{10}
$$
- Answer: $\boxed{\frac{3}{10}}$

---

Final Answers:

1. $\boxed{\frac{4}{9}}$
2. $\boxed{\frac{7}{10}}$
3. $\boxed{\frac{2}{9}}$
4. $\boxed{\frac{5}{6}}$
5. $\boxed{\frac{7}{12}}$
6. $\boxed{\frac{5}{8}}$
7. $\boxed{\frac{13}{24}}$
8. $\boxed{\frac{5}{12}}$
9. $\boxed{\frac{4}{21}}$
10. $\boxed{\frac{3}{10}}$