Math worksheet for adding fractions and reducing to lowest terms.

Problem: Add Fractions and Reduce to Lowest Terms

The task is to solve the given addition problems involving mixed numbers and fractions, then reduce the results to their lowest terms. Let's solve each problem step by step.

---

#### Problem 1: \( 9 \frac{6}{8} + \frac{8}{10} \)

1. Convert mixed number to an improper fraction:
\[
9 \frac{6}{8} = 9 + \frac{6}{8} = \frac{9 \times 8 + 6}{8} = \frac{72 + 6}{8} = \frac{78}{8}
\]

2. Simplify \(\frac{78}{8}\):
\[
\frac{78}{8} = \frac{39}{4}
\]

3. Add \(\frac{39}{4}\) and \(\frac{8}{10}\):
- Find a common denominator for 4 and 10. The least common denominator (LCD) is 20.
- Convert \(\frac{39}{4}\) to a fraction with denominator 20:
\[
\frac{39}{4} = \frac{39 \times 5}{4 \times 5} = \frac{195}{20}
\]
- Convert \(\frac{8}{10}\) to a fraction with denominator 20:
\[
\frac{8}{10} = \frac{8 \times 2}{10 \times 2} = \frac{16}{20}
\]
- Add the fractions:
\[
\frac{195}{20} + \frac{16}{20} = \frac{195 + 16}{20} = \frac{211}{20}
\]

4. Convert \(\frac{211}{20}\) back to a mixed number:
\[
\frac{211}{20} = 10 \frac{11}{20}
\]

5. Final Answer:
\[
\boxed{10 \frac{11}{20}}
\]

---

#### Problem 2: \( 1 \frac{2}{3} + \frac{2}{6} \)

1. Convert mixed number to an improper fraction:
\[
1 \frac{2}{3} = 1 + \frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3}
\]

2. Simplify \(\frac{2}{6}\):
\[
\frac{2}{6} = \frac{1}{3}
\]

3. Add \(\frac{5}{3}\) and \(\frac{1}{3}\):
\[
\frac{5}{3} + \frac{1}{3} = \frac{5 + 1}{3} = \frac{6}{3} = 2
\]

4. Final Answer:
\[
\boxed{2}
\]

---

#### Problem 3: \( 6 \frac{2}{3} + \frac{1}{3} \)

1. Convert mixed number to an improper fraction:
\[
6 \frac{2}{3} = 6 + \frac{2}{3} = \frac{6 \times 3 + 2}{3} = \frac{18 + 2}{3} = \frac{20}{3}
\]

2. Add \(\frac{20}{3}\) and \(\frac{1}{3}\):
\[
\frac{20}{3} + \frac{1}{3} = \frac{20 + 1}{3} = \frac{21}{3} = 7
\]

3. Final Answer:
\[
\boxed{7}
\]

---

#### Problem 4: \( 6 \frac{1}{5} + \frac{7}{8} \)

1. Convert mixed number to an improper fraction:
\[
6 \frac{1}{5} = 6 + \frac{1}{5} = \frac{6 \times 5 + 1}{5} = \frac{30 + 1}{5} = \frac{31}{5}
\]

2. Find a common denominator for 5 and 8. The LCD is 40.
- Convert \(\frac{31}{5}\) to a fraction with denominator 40:
\[
\frac{31}{5} = \frac{31 \times 8}{5 \times 8} = \frac{248}{40}
\]
- Convert \(\frac{7}{8}\) to a fraction with denominator 40:
\[
\frac{7}{8} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40}
\]
- Add the fractions:
\[
\frac{248}{40} + \frac{35}{40} = \frac{248 + 35}{40} = \frac{283}{40}
\]

3. Convert \(\frac{283}{40}\) back to a mixed number:
\[
\frac{283}{40} = 7 \frac{3}{40}
\]

4. Final Answer:
\[
\boxed{7 \frac{3}{40}}
\]

---

#### Problem 5: \( 4 \frac{2}{4} + \frac{3}{4} \)

1. Simplify \(\frac{2}{4}\):
\[
\frac{2}{4} = \frac{1}{2}
\]
So, \( 4 \frac{2}{4} = 4 \frac{1}{2} \).

2. Convert mixed number to an improper fraction:
\[
4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2}
\]

3. Add \(\frac{9}{2}\) and \(\frac{3}{4}\):
- Find a common denominator for 2 and 4. The LCD is 4.
- Convert \(\frac{9}{2}\) to a fraction with denominator 4:
\[
\frac{9}{2} = \frac{9 \times 2}{2 \times 2} = \frac{18}{4}
\]
- Add the fractions:
\[
\frac{18}{4} + \frac{3}{4} = \frac{18 + 3}{4} = \frac{21}{4}
\]

4. Convert \(\frac{21}{4}\) back to a mixed number:
\[
\frac{21}{4} = 5 \frac{1}{4}
\]

5. Final Answer:
\[
\boxed{5 \frac{1}{4}}
\]

---

#### Problem 6: \( 8 \frac{1}{5} + \frac{2}{4} \)

1. Simplify \(\frac{2}{4}\):
\[
\frac{2}{4} = \frac{1}{2}
\]

2. Convert mixed number to an improper fraction:
\[
8 \frac{1}{5} = 8 + \frac{1}{5} = \frac{8 \times 5 + 1}{5} = \frac{40 + 1}{5} = \frac{41}{5}
\]

3. Find a common denominator for 5 and 2. The LCD is 10.
- Convert \(\frac{41}{5}\) to a fraction with denominator 10:
\[
\frac{41}{5} = \frac{41 \times 2}{5 \times 2} = \frac{82}{10}
\]
- Convert \(\frac{1}{2}\) to a fraction with denominator 10:
\[
\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}
\]
- Add the fractions:
\[
\frac{82}{10} + \frac{5}{10} = \frac{82 + 5}{10} = \frac{87}{10}
\]

4. Convert \(\frac{87}{10}\) back to a mixed number:
\[
\frac{87}{10} = 8 \frac{7}{10}
\]

5. Final Answer:
\[
\boxed{8 \frac{7}{10}}
\]

---

#### Problem 7: \( 2 \frac{3}{8} + \frac{7}{8} \)

1. Convert mixed number to an improper fraction:
\[
2 \frac{3}{8} = 2 + \frac{3}{8} = \frac{2 \times 8 + 3}{8} = \frac{16 + 3}{8} = \frac{19}{8}
\]

2. Add \(\frac{19}{8}\) and \(\frac{7}{8}\):
\[
\frac{19}{8} + \frac{7}{8} = \frac{19 + 7}{8} = \frac{26}{8}
\]

3. Simplify \(\frac{26}{8}\):
\[
\frac{26}{8} = \frac{13}{4}
\]

4. Convert \(\frac{13}{4}\) back to a mixed number:
\[
\frac{13}{4} = 3 \frac{1}{4}
\]

5. Final Answer:
\[
\boxed{3 \frac{1}{4}}
\]

---

#### Problem 8: \( 9 \frac{3}{8} + \frac{1}{6} \)

1. Convert mixed number to an improper fraction:
\[
9 \frac{3}{8} = 9 + \frac{3}{8} = \frac{9 \times 8 + 3}{8} = \frac{72 + 3}{8} = \frac{75}{8}
\]

2. Find a common denominator for 8 and 6. The LCD is 24.
- Convert \(\frac{75}{8}\) to a fraction with denominator 24:
\[
\frac{75}{8} = \frac{75 \times 3}{8 \times 3} = \frac{225}{24}
\]
- Convert \(\frac{1}{6}\) to a fraction with denominator 24:
\[
\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}
\]
- Add the fractions:
\[
\frac{225}{24} + \frac{4}{24} = \frac{225 + 4}{24} = \frac{229}{24}
\]

3. Convert \(\frac{229}{24}\) back to a mixed number:
\[
\frac{229}{24} = 9 \frac{13}{24}
\]

4. Final Answer:
\[
\boxed{9 \frac{13}{24}}
\]

---

#### Problem 9: \( 8 \frac{2}{5} + \frac{1}{3} \)

1. Convert mixed number to an improper fraction:
\[
8 \frac{2}{5} = 8 + \frac{2}{5} = \frac{8 \times 5 + 2}{5} = \frac{40 + 2}{5} = \frac{42}{5}
\]

2. Find a common denominator for 5 and 3. The LCD is 15.
- Convert \(\frac{42}{5}\) to a fraction with denominator 15:
\[
\frac{42}{5} = \frac{42 \times 3}{5 \times 3} = \frac{126}{15}
\]
- Convert \(\frac{1}{3}\) to a fraction with denominator 15:
\[
\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}
\]
- Add the fractions:
\[
\frac{126}{15} + \frac{5}{15} = \frac{126 + 5}{15} = \frac{131}{15}
\]

3. Convert \(\frac{131}{15}\) back to a mixed number:
\[
\frac{131}{15} = 8 \frac{11}{15}
\]

4. Final Answer:
\[
\boxed{8 \frac{11}{15}}
\]

---

#### Problem 10: \( 7 \frac{4}{6} + \frac{4}{6} \)

1. Simplify \(\frac{4}{6}\):
\[
\frac{4}{6} = \frac{2}{3}
\]
So, \( 7 \frac{4}{6} = 7 \frac{2}{3} \).

2. Convert mixed number to an improper fraction:
\[
7 \frac{2}{3} = 7 + \frac{2}{3} = \frac{7 \times 3 + 2}{3} = \frac{21 + 2}{3} = \frac{23}{3}
\]

3. Add \(\frac{23}{3}\) and \(\frac{2}{3}\):
\[
\frac{23}{3} + \frac{2}{3} = \frac{23 + 2}{3} = \frac{25}{3}
\]

4. Convert \(\frac{25}{3}\) back to a mixed number:
\[
\frac{25}{3} = 8 \frac{1}{3}
\]

5. Final Answer:
\[
\boxed{8 \frac{1}{3}}
\]

---

#### Problem 11: \( 9 \frac{3}{8} + \frac{2}{8} \)

1. Convert mixed number to an improper fraction:
\[
9 \frac{3}{8} = 9 + \frac{3}{8} = \frac{9 \times 8 + 3}{8} = \frac{72 + 3}{8} = \frac{75}{8}
\]

2. Add \(\frac{75}{8}\) and \(\frac{2}{8}\):
\[
\frac{75}{8} + \frac{2}{8} = \frac{75 + 2}{8} = \frac{77}{8}
\]

3. Convert \(\frac{77}{8}\) back to a mixed number:
\[
\frac{77}{8} = 9 \frac{5}{8}
\]

4. Final Answer:
\[
\boxed{9 \frac{5}{8}}
\]

---

#### Problem 12: \( 8 \frac{2}{3} + \frac{1}{5} \)

1. Convert mixed number to an improper fraction:
\[
8 \frac{2}{3} = 8 + \frac{2}{3} = \frac{8 \times 3 + 2}{3} = \frac{24 + 2}{3} = \frac{26}{3}
\]

2. Find a common denominator for 3 and 5. The LCD is 15.
- Convert \(\frac{26}{3}\) to a fraction with denominator 15:
\[
\frac{26}{3} = \frac{26 \times 5}{3 \times 5} = \frac{130}{15}
\]
- Convert \(\frac{1}{5}\) to a fraction with denominator 15:
\[
\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}
\]
- Add the fractions:
\[
\frac{130}{15} + \frac{3}{15} = \frac{130 + 3}{15} = \frac{133}{15}
\]

3. Convert \(\frac{133}{15}\) back to a mixed number:
\[
\frac{133}{15} = 8 \frac{13}{15}
\]

4. Final Answer:
\[
\boxed{8 \frac{13}{15}}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 10 \frac{11}{20} \\
2. & \ 2 \\
3. & \ 7 \\
4. & \ 7 \frac{3}{40} \\
5. & \ 5 \frac{1}{4} \\
6. & \ 8 \frac{7}{10} \\
7. & \ 3 \frac{1}{4} \\
8. & \ 9 \frac{13}{24} \\
9. & \ 8 \frac{11}{15} \\
10. & \ 8 \frac{1}{3} \\
11. & \ 9 \frac{5}{8} \\
12. & \ 8 \frac{13}{15}
\end{aligned}
}
\]