Fraction arithmetic problems for practice.

Let's solve each problem step by step. We'll handle each expression one at a time.

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1. \( \left( \frac{3}{5} - \frac{2}{5} + \frac{1}{4} \right) \div \frac{1}{6} \)

#### Step 1: Simplify inside the parentheses
- First, simplify \( \frac{3}{5} - \frac{2}{5} \):
\[
\frac{3}{5} - \frac{2}{5} = \frac{3 - 2}{5} = \frac{1}{5}
\]
- Now add \( \frac{1}{4} \):
\[
\frac{1}{5} + \frac{1}{4}
\]
To add these fractions, find a common denominator. The least common multiple of 5 and 4 is 20:
\[
\frac{1}{5} = \frac{4}{20}, \quad \frac{1}{4} = \frac{5}{20}
\]
So:
\[
\frac{1}{5} + \frac{1}{4} = \frac{4}{20} + \frac{5}{20} = \frac{9}{20}
\]

#### Step 2: Divide by \( \frac{1}{6} \)
- Dividing by \( \frac{1}{6} \) is the same as multiplying by 6:
\[
\frac{9}{20} \div \frac{1}{6} = \frac{9}{20} \times 6 = \frac{9 \times 6}{20} = \frac{54}{20}
\]
- Simplify \( \frac{54}{20} \):
\[
\frac{54}{20} = \frac{27}{10}
\]

#### Final Answer:
\[
\boxed{\frac{27}{10}}
\]

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2. \( \left( \frac{1}{4} + \frac{3}{4} - \frac{3}{8} \right) \div \frac{1}{9} \)

#### Step 1: Simplify inside the parentheses
- First, simplify \( \frac{1}{4} + \frac{3}{4} \):
\[
\frac{1}{4} + \frac{3}{4} = \frac{1 + 3}{4} = \frac{4}{4} = 1
\]
- Now subtract \( \frac{3}{8} \):
\[
1 - \frac{3}{8}
\]
Write 1 as \( \frac{8}{8} \):
\[
1 - \frac{3}{8} = \frac{8}{8} - \frac{3}{8} = \frac{5}{8}
\]

#### Step 2: Divide by \( \frac{1}{9} \)
- Dividing by \( \frac{1}{9} \) is the same as multiplying by 9:
\[
\frac{5}{8} \div \frac{1}{9} = \frac{5}{8} \times 9 = \frac{5 \times 9}{8} = \frac{45}{8}
\]

#### Final Answer:
\[
\boxed{\frac{45}{8}}
\]

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3. \( \left( \frac{3}{8} \times \frac{1}{4} + \frac{7}{8} \right) \div \frac{1}{3} \)

#### Step 1: Simplify inside the parentheses
- First, calculate \( \frac{3}{8} \times \frac{1}{4} \):
\[
\frac{3}{8} \times \frac{1}{4} = \frac{3 \times 1}{8 \times 4} = \frac{3}{32}
\]
- Now add \( \frac{7}{8} \):
\[
\frac{3}{32} + \frac{7}{8}
\]
To add these fractions, find a common denominator. The least common multiple of 32 and 8 is 32:
\[
\frac{7}{8} = \frac{7 \times 4}{8 \times 4} = \frac{28}{32}
\]
So:
\[
\frac{3}{32} + \frac{7}{8} = \frac{3}{32} + \frac{28}{32} = \frac{31}{32}
\]

#### Step 2: Divide by \( \frac{1}{3} \)
- Dividing by \( \frac{1}{3} \) is the same as multiplying by 3:
\[
\frac{31}{32} \div \frac{1}{3} = \frac{31}{32} \times 3 = \frac{31 \times 3}{32} = \frac{93}{32}
\]

#### Final Answer:
\[
\boxed{\frac{93}{32}}
\]

---

4. \( \left( \frac{8}{9} - \frac{2}{3} + \frac{5}{8} \right) \div \frac{1}{3} \)

#### Step 1: Simplify inside the parentheses
- First, simplify \( \frac{8}{9} - \frac{2}{3} \):
\[
\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}
\]
So:
\[
\frac{8}{9} - \frac{2}{3} = \frac{8}{9} - \frac{6}{9} = \frac{2}{9}
\]
- Now add \( \frac{5}{8} \):
\[
\frac{2}{9} + \frac{5}{8}
\]
To add these fractions, find a common denominator. The least common multiple of 9 and 8 is 72:
\[
\frac{2}{9} = \frac{2 \times 8}{9 \times 8} = \frac{16}{72}, \quad \frac{5}{8} = \frac{5 \times 9}{8 \times 9} = \frac{45}{72}
\]
So:
\[
\frac{2}{9} + \frac{5}{8} = \frac{16}{72} + \frac{45}{72} = \frac{61}{72}
\]

#### Step 2: Divide by \( \frac{1}{3} \)
- Dividing by \( \frac{1}{3} \) is the same as multiplying by 3:
\[
\frac{61}{72} \div \frac{1}{3} = \frac{61}{72} \times 3 = \frac{61 \times 3}{72} = \frac{183}{72}
\]
- Simplify \( \frac{183}{72} \):
\[
\frac{183}{72} = \frac{61}{24}
\]

#### Final Answer:
\[
\boxed{\frac{61}{24}}
\]

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5. \( \left( \frac{1}{4} + \frac{1}{8} - \frac{1}{5} \right) \times \frac{4}{9} \)

#### Step 1: Simplify inside the parentheses
- First, simplify \( \frac{1}{4} + \frac{1}{8} \):
\[
\frac{1}{4} = \frac{2}{8}, \quad \text{so: } \frac{1}{4} + \frac{1}{8} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8}
\]
- Now subtract \( \frac{1}{5} \):
\[
\frac{3}{8} - \frac{1}{5}
\]
To subtract these fractions, find a common denominator. The least common multiple of 8 and 5 is 40:
\[
\frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40}, \quad \frac{1}{5} = \frac{1 \times 8}{5 \times 8} = \frac{8}{40}
\]
So:
\[
\frac{3}{8} - \frac{1}{5} = \frac{15}{40} - \frac{8}{40} = \frac{7}{40}
\]

#### Step 2: Multiply by \( \frac{4}{9} \)
- Multiply \( \frac{7}{40} \) by \( \frac{4}{9} \):
\[
\frac{7}{40} \times \frac{4}{9} = \frac{7 \times 4}{40 \times 9} = \frac{28}{360}
\]
- Simplify \( \frac{28}{360} \):
\[
\frac{28}{360} = \frac{7}{90}
\]

#### Final Answer:
\[
\boxed{\frac{7}{90}}
\]

---

6. \( \frac{1}{8} \div \left( \frac{3}{5} + \frac{5}{6} - \frac{1}{3} \right) \)

#### Step 1: Simplify inside the parentheses
- First, simplify \( \frac{3}{5} + \frac{5}{6} \):
\[
\text{Least common multiple of 5 and 6 is 30: }
\]
\[
\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}, \quad \frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}
\]
So:
\[
\frac{3}{5} + \frac{5}{6} = \frac{18}{30} + \frac{25}{30} = \frac{43}{30}
\]
- Now subtract \( \frac{1}{3} \):
\[
\frac{43}{30} - \frac{1}{3}
\]
Write \( \frac{1}{3} \) with a denominator of 30:
\[
\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}
\]
So:
\[
\frac{43}{30} - \frac{1}{3} = \frac{43}{30} - \frac{10}{30} = \frac{33}{30}
\]
- Simplify \( \frac{33}{30} \):
\[
\frac{33}{30} = \frac{11}{10}
\]

#### Step 2: Divide \( \frac{1}{8} \) by \( \frac{11}{10} \)
- Dividing by \( \frac{11}{10} \) is the same as multiplying by \( \frac{10}{11} \):
\[
\frac{1}{8} \div \frac{11}{10} = \frac{1}{8} \times \frac{10}{11} = \frac{1 \times 10}{8 \times 11} = \frac{10}{88}
\]
- Simplify \( \frac{10}{88} \):
\[
\frac{10}{88} = \frac{5}{44}
\]

#### Final Answer:
\[
\boxed{\frac{5}{44}}
\]

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7. \( \left( \frac{7}{9} + \frac{5}{6} - \frac{2}{3} \right) \times \frac{1}{3} \)

#### Step 1: Simplify inside the parentheses
- First, simplify \( \frac{7}{9} + \frac{5}{6} \):
\[
\text{Least common multiple of 9 and 6 is 18: }
\]
\[
\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}, \quad \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}
\]
So:
\[
\frac{7}{9} + \frac{5}{6} = \frac{14}{18} + \frac{15}{18} = \frac{29}{18}
\]
- Now subtract \( \frac{2}{3} \):
\[
\frac{29}{18} - \frac{2}{3}
\]
Write \( \frac{2}{3} \) with a denominator of 18:
\[
\frac{2}{3} = \frac{2 \times 6}{3 \times 6} = \frac{12}{18}
\]
So:
\[
\frac{29}{18} - \frac{2}{3} = \frac{29}{18} - \frac{12}{18} = \frac{17}{18}
\]

#### Step 2: Multiply by \( \frac{1}{3} \)
- Multiply \( \frac{17}{18} \) by \( \frac{1}{3} \):
\[
\frac{17}{18} \times \frac{1}{3} = \frac{17 \times 1}{18 \times 3} = \frac{17}{54}
\]

#### Final Answer:
\[
\boxed{\frac{17}{54}}
\]

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8. \( \left( \frac{2}{9} + \frac{1}{9} \right) \times \left( \frac{1}{3} - \frac{1}{4} \right) \)

#### Step 1: Simplify each part inside the parentheses
- Simplify \( \frac{2}{9} + \frac{1}{9} \):
\[
\frac{2}{9} + \frac{1}{9} = \frac{2 + 1}{9} = \frac{3}{9} = \frac{1}{3}
\]
- Simplify \( \frac{1}{3} - \frac{1}{4} \):
\[
\text{Least common multiple of 3 and 4 is 12: }
\]
\[
\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}, \quad \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}
\]
So:
\[
\frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}
\]

#### Step 2: Multiply the results
- Multiply \( \frac{1}{3} \) by \( \frac{1}{12} \):
\[
\frac{1}{3} \times \frac{1}{12} = \frac{1 \times 1}{3 \times 12} = \frac{1}{36}
\]

#### Final Answer:
\[
\boxed{\frac{1}{36}}
\]

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9. \( \left( \frac{1}{6} \times \left( \frac{4}{5} - \frac{1}{3} \right) \right) \div \frac{1}{5} \)

#### Step 1: Simplify inside the inner parentheses
- Simplify \( \frac{4}{5} - \frac{1}{3} \):
\[
\text{Least common multiple of 5 and 3 is 15: }
\]
\[
\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}, \quad \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}
\]
So:
\[
\frac{4}{5} - \frac{1}{3} = \frac{12}{15} - \frac{5}{15} = \frac{7}{15}
\]

#### Step 2: Multiply \( \frac{1}{6} \) by \( \frac{7}{15} \)
- Multiply \( \frac{1}{6} \) by \( \frac{7}{15} \):
\[
\frac{1}{6} \times \frac{7}{15} = \frac{1 \times 7}{6 \times 15} = \frac{7}{90}
\]

#### Step 3: Divide by \( \frac{1}{5} \)
- Dividing by \( \frac{1}{5} \) is the same as multiplying by 5:
\[
\frac{7}{90} \div \frac{1}{5} = \frac{7}{90} \times 5 = \frac{7 \times 5}{90} = \frac{35}{90}
\]
- Simplify \( \frac{35}{90} \):
\[
\frac{35}{90} = \frac{7}{18}
\]

#### Final Answer:
\[
\boxed{\frac{7}{18}}
\]

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Final Answers:

\[
\boxed{\frac{27}{10}, \frac{45}{8}, \frac{93}{32}, \frac{61}{24}, \frac{7}{90}, \frac{5}{44}, \frac{17}{54}, \frac{1}{36}, \frac{7}{18}}
\]