Operations with Two Mixed Fractions with Unlike Denominators ... - Free Printable

The image shows a worksheet titled "Operations with Two Mixed Fractions (A) Answers," which provides solutions to various problems involving mixed fractions. Below, I will explain the solution process for each problem step by step.

---

Problem 1: \( 5 \frac{2}{8} - 1 \frac{2}{7} \)

#### Solution:
1. Convert mixed fractions to improper fractions:
\[
5 \frac{2}{8} = \frac{5 \times 8 + 2}{8} = \frac{40 + 2}{8} = \frac{42}{8}
\]
\[
1 \frac{2}{7} = \frac{1 \times 7 + 2}{7} = \frac{7 + 2}{7} = \frac{9}{7}
\]

2. Find a common denominator for \( \frac{42}{8} \) and \( \frac{9}{7} \):
The least common denominator (LCD) of 8 and 7 is 56.

3. Rewrite the fractions with the common denominator:
\[
\frac{42}{8} = \frac{42 \times 7}{8 \times 7} = \frac{294}{56}
\]
\[
\frac{9}{7} = \frac{9 \times 8}{7 \times 8} = \frac{72}{56}
\]

4. Subtract the fractions:
\[
\frac{294}{56} - \frac{72}{56} = \frac{294 - 72}{56} = \frac{222}{56}
\]

5. Simplify the fraction:
\[
\frac{222}{56} = \frac{111}{28} \quad (\text{dividing numerator and denominator by 2})
\]

6. Convert back to a mixed fraction:
\[
\frac{111}{28} = 3 \frac{27}{28} \quad (\text{since } 111 \div 28 = 3 \text{ remainder } 27)
\]

Final Answer:
\[
\boxed{3 \frac{27}{28}}
\]

---

Problem 2: \( 5 \frac{1}{2} - 5 \frac{2}{5} \)

#### Solution:
1. Convert mixed fractions to improper fractions:
\[
5 \frac{1}{2} = \frac{5 \times 2 + 1}{2} = \frac{10 + 1}{2} = \frac{11}{2}
\]
\[
5 \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{25 + 2}{5} = \frac{27}{5}
\]

2. Find a common denominator for \( \frac{11}{2} \) and \( \frac{27}{5} \):
The least common denominator (LCD) of 2 and 5 is 10.

3. Rewrite the fractions with the common denominator:
\[
\frac{11}{2} = \frac{11 \times 5}{2 \times 5} = \frac{55}{10}
\]
\[
\frac{27}{5} = \frac{27 \times 2}{5 \times 2} = \frac{54}{10}
\]

4. Subtract the fractions:
\[
\frac{55}{10} - \frac{54}{10} = \frac{55 - 54}{10} = \frac{1}{10}
\]

Final Answer:
\[
\boxed{\frac{1}{10}}
\]

---

Problem 3: \( 5 \frac{4}{8} \times 1 \frac{5}{14} \)

#### Solution:
1. Convert mixed fractions to improper fractions:
\[
5 \frac{4}{8} = \frac{5 \times 8 + 4}{8} = \frac{40 + 4}{8} = \frac{44}{8}
\]
\[
1 \frac{5}{14} = \frac{1 \times 14 + 5}{14} = \frac{14 + 5}{14} = \frac{19}{14}
\]

2. Multiply the fractions:
\[
\frac{44}{8} \times \frac{19}{14} = \frac{44 \times 19}{8 \times 14} = \frac{836}{112}
\]

3. Simplify the fraction:
\[
\frac{836}{112} = \frac{209}{28} \quad (\text{dividing numerator and denominator by 4})
\]

4. Convert back to a mixed fraction:
\[
\frac{209}{28} = 7 \frac{13}{28} \quad (\text{since } 209 \div 28 = 7 \text{ remainder } 13)
\]

Final Answer:
\[
\boxed{7 \frac{13}{28}}
\]

---

Problem 4: \( 5 \frac{3}{4} \div 2 \frac{3}{16} \)

#### Solution:
1. Convert mixed fractions to improper fractions:
\[
5 \frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4}
\]
\[
2 \frac{3}{16} = \frac{2 \times 16 + 3}{16} = \frac{32 + 3}{16} = \frac{35}{16}
\]

2. Divide the fractions by multiplying by the reciprocal:
\[
\frac{23}{4} \div \frac{35}{16} = \frac{23}{4} \times \frac{16}{35} = \frac{23 \times 16}{4 \times 35} = \frac{368}{140}
\]

3. Simplify the fraction:
\[
\frac{368}{140} = \frac{92}{35} \quad (\text{dividing numerator and denominator by 4})
\]

4. Convert back to a mixed fraction:
\[
\frac{92}{35} = 2 \frac{22}{35} \quad (\text{since } 92 \div 35 = 2 \text{ remainder } 22)
\]

Final Answer:
\[
\boxed{2 \frac{22}{35}}
\]

---

Problem 5: \( 1 \frac{2}{19} \times 5 \frac{2}{3} \)

#### Solution:
1. Convert mixed fractions to improper fractions:
\[
1 \frac{2}{19} = \frac{1 \times 19 + 2}{19} = \frac{19 + 2}{19} = \frac{21}{19}
\]
\[
5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3}
\]

2. Multiply the fractions:
\[
\frac{21}{19} \times \frac{17}{3} = \frac{21 \times 17}{19 \times 3} = \frac{357}{57}
\]

3. Simplify the fraction:
\[
\frac{357}{57} = \frac{119}{19} \quad (\text{dividing numerator and denominator by 3})
\]

4. Convert back to a mixed fraction:
\[
\frac{119}{19} = 6 \frac{5}{19} \quad (\text{since } 119 \div 19 = 6 \text{ remainder } 5)
\]

Final Answer:
\[
\boxed{6 \frac{5}{19}}
\]

---

Problem 6: \( 5 \frac{1}{2} + 2 \frac{6}{9} \)

#### Solution:
1. Convert mixed fractions to improper fractions:
\[
5 \frac{1}{2} = \frac{5 \times 2 + 1}{2} = \frac{10 + 1}{2} = \frac{11}{2}
\]
\[
2 \frac{6}{9} = \frac{2 \times 9 + 6}{9} = \frac{18 + 6}{9} = \frac{24}{9}
\]

2. Simplify \( \frac{24}{9} \):
\[
\frac{24}{9} = \frac{8}{3} \quad (\text{dividing numerator and denominator by 3})
\]

3. Find a common denominator for \( \frac{11}{2} \) and \( \frac{8}{3} \):
The least common denominator (LCD) of 2 and 3 is 6.

4. Rewrite the fractions with the common denominator:
\[
\frac{11}{2} = \frac{11 \times 3}{2 \times 3} = \frac{33}{6}
\]
\[
\frac{8}{3} = \frac{8 \times 2}{3 \times 2} = \frac{16}{6}
\]

5. Add the fractions:
\[
\frac{33}{6} + \frac{16}{6} = \frac{33 + 16}{6} = \frac{49}{6}
\]

6. Convert back to a mixed fraction:
\[
\frac{49}{6} = 8 \frac{1}{6} \quad (\text{since } 49 \div 6 = 8 \text{ remainder } 1)
\]

Final Answer:
\[
\boxed{8 \frac{1}{6}}
\]

---

Problem 7: \( 5 \frac{2}{3} \div 1 \frac{14}{20} \)

#### Solution:
1. Convert mixed fractions to improper fractions:
\[
5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3}
\]
\[
1 \frac{14}{20} = \frac{1 \times 20 + 14}{20} = \frac{20 + 14}{20} = \frac{34}{20}
\]

2. Simplify \( \frac{34}{20} \):
\[
\frac{34}{20} = \frac{17}{10} \quad (\text{dividing numerator and denominator by 2})
\]

3. Divide the fractions by multiplying by the reciprocal:
\[
\frac{17}{3} \div \frac{17}{10} = \frac{17}{3} \times \frac{10}{17} = \frac{17 \times 10}{3 \times 17} = \frac{170}{51}
\]

4. Simplify the fraction:
\[
\frac{170}{51} = \frac{34}{10} = \frac{17}{5} \quad (\text{dividing numerator and denominator by 2})
\]

5. Convert back to a mixed fraction:
\[
\frac{17}{5} = 3 \frac{2}{5} \quad (\text{since } 17 \div 5 = 3 \text{ remainder } 2)
\]

Final Answer:
\[
\boxed{3 \frac{2}{5}}
\]

---

Problem 8: \( 7 \frac{1}{12} \times 3 \frac{3}{5} \)

#### Solution:
1. Convert mixed fractions to improper fractions:
\[
7 \frac{1}{12} = \frac{7 \times 12 + 1}{12} = \frac{84 + 1}{12} = \frac{85}{12}
\]
\[
3 \frac{3}{5} = \frac{3 \times 5 + 3}{5} = \frac{15 + 3}{5} = \frac{18}{5}
\]

2. Multiply the fractions:
\[
\frac{85}{12} \times \frac{18}{5} = \frac{85 \times 18}{12 \times 5} = \frac{1530}{60}
\]

3. Simplify the fraction:
\[
\frac{1530}{60} = \frac{510}{20} = \frac{255}{10} = \frac{51}{2} \quad (\text{dividing numerator and denominator by 30})
\]

4. Convert back to a mixed fraction:
\[
\frac{51}{2} = 25 \frac{1}{2} \quad (\text{since } 51 \div 2 = 25 \text{ remainder } 1)
\]

Final Answer:
\[
\boxed{25 \frac{1}{2}}
\]

---

Problem 9: \( 5 \frac{3}{4} - 1 \frac{2}{9} \)

#### Solution:
1. Convert mixed fractions to improper fractions:
\[
5 \frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4}
\]
\[
1 \frac{2}{9} = \frac{1 \times 9 + 2}{9} = \frac{9 + 2}{9} = \frac{11}{9}
\]

2. Find a common denominator for \( \frac{23}{4} \) and \( \frac{11}{9} \):
The least common denominator (LCD) of 4 and 9 is 36.

3. Rewrite the fractions with the common denominator:
\[
\frac{23}{4} = \frac{23 \times 9}{4 \times 9} = \frac{207}{36}
\]
\[
\frac{11}{9} = \frac{11 \times 4}{9 \times 4} = \frac{44}{36}
\]

4. Subtract the fractions:
\[
\frac{207}{36} - \frac{44}{36} = \frac{207 - 44}{36} = \frac{163}{36}
\]

5. Convert back to a mixed fraction:
\[
\frac{163}{36} = 4 \frac{19}{36} \quad (\text{since } 163 \div 36 = 4 \text{ remainder } 19)
\]

Final Answer:
\[
\boxed{4 \frac{19}{36}}
\]

---

Problem 10: \( 1 \frac{13}{14} \div 5 \frac{5}{6} \)

#### Solution:
1. Convert mixed fractions to improper fractions:
\[
1 \frac{13}{14} = \frac{1 \times 14 + 13}{14} = \frac{14 + 13}{14} = \frac{27}{14}
\]
\[
5 \frac{5}{6} = \frac{5 \times 6 + 5}{6} = \frac{30 + 5}{6} = \frac{35}{6}
\]

2. Divide the fractions by multiplying by the reciprocal:
\[
\frac{27}{14} \div \frac{35}{6} = \frac{27}{14} \times \frac{6}{35} = \frac{27 \times 6}{14 \times 35} = \frac{162}{490}
\]

3. Simplify the fraction:
\[
\frac{162}{490} = \frac{81}{245} \quad (\text{dividing numerator and denominator by 2})
\]

Final Answer:
\[
\boxed{\frac{81}{245}}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 3 \frac{27}{28} \\
2. & \ \frac{1}{10} \\
3. & \ 7 \frac{13}{28} \\
4. & \ 2 \frac{22}{35} \\
5. & \ 6 \frac{5}{19} \\
6. & \ 8 \frac{1}{6} \\
7. & \ 3 \frac{2}{5} \\
8. & \ 25 \frac{1}{2} \\
9. & \ 4 \frac{19}{36} \\
10. & \ \frac{81}{245}
\end{aligned}
}
\]