Fraction arithmetic practice problems.

Problem: Solve the given arithmetic problems involving fractions.

#### Problem 67:
$$
\frac{4}{5} \div 6
$$

Solution:
1. Dividing by a number is the same as multiplying by its reciprocal.
$$
\frac{4}{5} \div 6 = \frac{4}{5} \times \frac{1}{6}
$$
2. Multiply the numerators and the denominators:
$$
\frac{4 \times 1}{5 \times 6} = \frac{4}{30}
$$
3. Simplify the fraction:
$$
\frac{4}{30} = \frac{2}{15}
$$

Answer:
$$
\boxed{\frac{2}{15}}
$$

---

#### Problem 68:
$$
\frac{8}{5} \div \frac{12}{16}
$$

Solution:
1. Dividing by a fraction is the same as multiplying by its reciprocal.
$$
\frac{8}{5} \div \frac{12}{16} = \frac{8}{5} \times \frac{16}{12}
$$
2. Simplify the second fraction:
$$
\frac{12}{16} = \frac{3}{4}
$$
So, the expression becomes:
$$
\frac{8}{5} \times \frac{16}{12} = \frac{8}{5} \times \frac{4}{3}
$$
3. Multiply the numerators and the denominators:
$$
\frac{8 \times 4}{5 \times 3} = \frac{32}{15}
$$

Answer:
$$
\boxed{\frac{32}{15}}
$$

---

#### Problem 69:
$$
\frac{15}{6} \div \frac{16}{5}
$$

Solution:
1. Dividing by a fraction is the same as multiplying by its reciprocal.
$$
\frac{15}{6} \div \frac{16}{5} = \frac{15}{6} \times \frac{5}{16}
$$
2. Simplify the first fraction:
$$
\frac{15}{6} = \frac{5}{2}
$$
So, the expression becomes:
$$
\frac{5}{2} \times \frac{5}{16}
$$
3. Multiply the numerators and the denominators:
$$
\frac{5 \times 5}{2 \times 16} = \frac{25}{32}
$$

Answer:
$$
\boxed{\frac{25}{32}}
$$

---

#### Problem 70:
$$
\frac{21}{4} \div \frac{7}{4}
$$

Solution:
1. Dividing by a fraction is the same as multiplying by its reciprocal.
$$
\frac{21}{4} \div \frac{7}{4} = \frac{21}{4} \times \frac{4}{7}
$$
2. Multiply the numerators and the denominators:
$$
\frac{21 \times 4}{4 \times 7} = \frac{84}{28}
$$
3. Simplify the fraction:
$$
\frac{84}{28} = 3
$$

Answer:
$$
\boxed{3}
$$

---

#### Problem 71:
$$
3 + \frac{1}{4} \div \frac{2}{3}
$$

Solution:
1. First, handle the division:
$$
\frac{1}{4} \div \frac{2}{3} = \frac{1}{4} \times \frac{3}{2} = \frac{1 \times 3}{4 \times 2} = \frac{3}{8}
$$
2. Now add the result to 3:
$$
3 + \frac{3}{8}
$$
3. Convert 3 to a fraction with a denominator of 8:
$$
3 = \frac{24}{8}
$$
4. Add the fractions:
$$
\frac{24}{8} + \frac{3}{8} = \frac{27}{8}
$$

Answer:
$$
\boxed{\frac{27}{8}}
$$

---

#### Problem 72:
$$
4 + \frac{1}{2} \div \frac{3}{5}
$$

Solution:
1. First, handle the division:
$$
\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \times \frac{5}{3} = \frac{1 \times 5}{2 \times 3} = \frac{5}{6}
$$
2. Now add the result to 4:
$$
4 + \frac{5}{6}
$$
3. Convert 4 to a fraction with a denominator of 6:
$$
4 = \frac{24}{6}
$$
4. Add the fractions:
$$
\frac{24}{6} + \frac{5}{6} = \frac{29}{6}
$$

Answer:
$$
\boxed{\frac{29}{6}}
$$

---

#### Problem 73:
$$
5 - \frac{3}{4} \div \frac{5}{6}
$$

Solution:
1. First, handle the division:
$$
\frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5} = \frac{3 \times 6}{4 \times 5} = \frac{18}{20}
$$
2. Simplify the fraction:
$$
\frac{18}{20} = \frac{9}{10}
$$
3. Now subtract the result from 5:
$$
5 - \frac{9}{10}
$$
4. Convert 5 to a fraction with a denominator of 10:
$$
5 = \frac{50}{10}
$$
5. Subtract the fractions:
$$
\frac{50}{10} - \frac{9}{10} = \frac{41}{10}
$$

Answer:
$$
\boxed{\frac{41}{10}}
$$

---

#### Problem 74:
$$
4 - \frac{3}{5} \div \frac{7}{10}
$$

Solution:
1. First, handle the division:
$$
\frac{3}{5} \div \frac{7}{10} = \frac{3}{5} \times \frac{10}{7} = \frac{3 \times 10}{5 \times 7} = \frac{30}{35}
$$
2. Simplify the fraction:
$$
\frac{30}{35} = \frac{6}{7}
$$
3. Now subtract the result from 4:
$$
4 - \frac{6}{7}
$$
4. Convert 4 to a fraction with a denominator of 7:
$$
4 = \frac{28}{7}
$$
5. Subtract the fractions:
$$
\frac{28}{7} - \frac{6}{7} = \frac{22}{7}
$$

Answer:
$$
\boxed{\frac{22}{7}}
$$

---

#### Problem 75:
$$
\frac{\frac{2}{3} + \frac{3}{4}}{4 - \frac{1}{3}}
$$

Solution:
1. Simplify the numerator:
$$
\frac{2}{3} + \frac{3}{4}
$$
Find a common denominator (12):
$$
\frac{2}{3} = \frac{8}{12}, \quad \frac{3}{4} = \frac{9}{12}
$$
Add the fractions:
$$
\frac{8}{12} + \frac{9}{12} = \frac{17}{12}
$$
2. Simplify the denominator:
$$
4 - \frac{1}{3}
$$
Convert 4 to a fraction with a denominator of 3:
$$
4 = \frac{12}{3}
$$
Subtract the fractions:
$$
\frac{12}{3} - \frac{1}{3} = \frac{11}{3}
$$
3. Now divide the simplified numerator by the simplified denominator:
$$
\frac{\frac{17}{12}}{\frac{11}{3}} = \frac{17}{12} \times \frac{3}{11} = \frac{17 \times 3}{12 \times 11} = \frac{51}{132}
$$
4. Simplify the fraction:
$$
\frac{51}{132} = \frac{17}{44}
$$

Answer:
$$
\boxed{\frac{17}{44}}
$$

---

#### Problem 76:
$$
\frac{\frac{5}{6} + \frac{2}{3}}{3 - \frac{1}{2}}
$$

Solution:
1. Simplify the numerator:
$$
\frac{5}{6} + \frac{2}{3}
$$
Find a common denominator (6):
$$
\frac{5}{6}, \quad \frac{2}{3} = \frac{4}{6}
$$
Add the fractions:
$$
\frac{5}{6} + \frac{4}{6} = \frac{9}{6} = \frac{3}{2}
$$
2. Simplify the denominator:
$$
3 - \frac{1}{2}
$$
Convert 3 to a fraction with a denominator of 2:
$$
3 = \frac{6}{2}
$$
Subtract the fractions:
$$
\frac{6}{2} - \frac{1}{2} = \frac{5}{2}
$$
3. Now divide the simplified numerator by the simplified denominator:
$$
\frac{\frac{3}{2}}{\frac{5}{2}} = \frac{3}{2} \times \frac{2}{5} = \frac{3 \times 2}{2 \times 5} = \frac{6}{10} = \frac{3}{5}
$$

Answer:
$$
\boxed{\frac{3}{5}}
$$

---

#### Problem 77:
$$
\frac{\frac{4}{5} - \frac{3}{6}}{\frac{1}{5} + \frac{1}{2}}
$$

Solution:
1. Simplify the numerator:
$$
\frac{4}{5} - \frac{3}{6}
$$
Simplify $\frac{3}{6}$:
$$
\frac{3}{6} = \frac{1}{2}
$$
Find a common denominator (10):
$$
\frac{4}{5} = \frac{8}{10}, \quad \frac{1}{2} = \frac{5}{10}
$$
Subtract the fractions:
$$
\frac{8}{10} - \frac{5}{10} = \frac{3}{10}
$$
2. Simplify the denominator:
$$
\frac{1}{5} + \frac{1}{2}
$$
Find a common denominator (10):
$$
\frac{1}{5} = \frac{2}{10}, \quad \frac{1}{2} = \frac{5}{10}
$$
Add the fractions:
$$
\frac{2}{10} + \frac{5}{10} = \frac{7}{10}
$$
3. Now divide the simplified numerator by the simplified denominator:
$$
\frac{\frac{3}{10}}{\frac{7}{10}} = \frac{3}{10} \times \frac{10}{7} = \frac{3 \times 10}{10 \times 7} = \frac{30}{70} = \frac{3}{7}
$$

Answer:
$$
\boxed{\frac{3}{7}}
$$

---

#### Problem 78:
$$
\frac{\frac{2}{3} + \frac{1}{6}}{\frac{3}{4} - \frac{5}{8}}
$$

Solution:
1. Simplify the numerator:
$$
\frac{2}{3} + \frac{1}{6}
$$
Find a common denominator (6):
$$
\frac{2}{3} = \frac{4}{6}, \quad \frac{1}{6}
$$
Add the fractions:
$$
\frac{4}{6} + \frac{1}{6} = \frac{5}{6}
$$
2. Simplify the denominator:
$$
\frac{3}{4} - \frac{5}{8}
$$
Find a common denominator (8):
$$
\frac{3}{4} = \frac{6}{8}, \quad \frac{5}{8}
$$
Subtract the fractions:
$$
\frac{6}{8} - \frac{5}{8} = \frac{1}{8}
$$
3. Now divide the simplified numerator by the simplified denominator:
$$
\frac{\frac{5}{6}}{\frac{1}{8}} = \frac{5}{6} \times \frac{8}{1} = \frac{5 \times 8}{6 \times 1} = \frac{40}{6} = \frac{20}{3}
$$

Answer:
$$
\boxed{\frac{20}{3}}
$$

---

#### Problem 79:
$$
\frac{1 \frac{1}{2} - 2 \frac{2}{3}}{3 \frac{1}{4} + 1 \frac{1}{6}}
$$

Solution:
1. Convert mixed numbers to improper fractions:
$$
1 \frac{1}{2} = \frac{3}{2}, \quad 2 \frac{2}{3} = \frac{8}{3}, \quad 3 \frac{1}{4} = \frac{13}{4}, \quad 1 \frac{1}{6} = \frac{7}{6}
$$
2. Simplify the numerator:
$$
\frac{3}{2} - \frac{8}{3}
$$
Find a common denominator (6):
$$
\frac{3}{2} = \frac{9}{6}, \quad \frac{8}{3} = \frac{16}{6}
$$
Subtract the fractions:
$$
\frac{9}{6} - \frac{16}{6} = \frac{-7}{6}
$$
3. Simplify the denominator:
$$
\frac{13}{4} + \frac{7}{6}
$$
Find a common denominator (12):
$$
\frac{13}{4} = \frac{39}{12}, \quad \frac{7}{6} = \frac{14}{12}
$$
Add the fractions:
$$
\frac{39}{12} + \frac{14}{12} = \frac{53}{12}
$$
4. Now divide the simplified numerator by the simplified denominator:
$$
\frac{\frac{-7}{6}}{\frac{53}{12}} = \frac{-7}{6} \times \frac{12}{53} = \frac{-7 \times 12}{6 \times 53} = \frac{-84}{318} = \frac{-14}{53}
$$

Answer:
$$
\boxed{\frac{-14}{53}}
$$

---

#### Problem 80:
$$
\frac{1 \frac{1}{3} - 4 \frac{1}{2}}{2 \frac{1}{4} + 3 \frac{1}{8}}
$$

Solution:
1. Convert mixed numbers to improper fractions:
$$
1 \frac{1}{3} = \frac{4}{3}, \quad 4 \frac{1}{2} = \frac{9}{2}, \quad 2 \frac{1}{4} = \frac{9}{4}, \quad 3 \frac{1}{8} = \frac{25}{8}
$$
2. Simplify the numerator:
$$
\frac{4}{3} - \frac{9}{2}
$$
Find a common denominator (6):
$$
\frac{4}{3} = \frac{8}{6}, \quad \frac{9}{2} = \frac{27}{6}
$$
Subtract the fractions:
$$
\frac{8}{6} - \frac{27}{6} = \frac{-19}{6}
$$
3. Simplify the denominator:
$$
\frac{9}{4} + \frac{25}{8}
$$
Find a common denominator (8):
$$
\frac{9}{4} = \frac{18}{8}, \quad \frac{25}{8}
$$
Add the fractions:
$$
\frac{18}{8} + \frac{25}{8} = \frac{43}{8}
$$
4. Now divide the simplified numerator by the simplified denominator:
$$
\frac{\frac{-19}{6}}{\frac{43}{8}} = \frac{-19}{6} \times \frac{8}{43} = \frac{-19 \times 8}{6 \times 43} = \frac{-152}{258} = \frac{-76}{129}
$$

Answer:
$$
\boxed{\frac{-76}{129}}
$$

---

#### Problem 81:
$$
\frac{2 \frac{1}{6} + 4 \frac{2}{3}}{2 \frac{1}{3} - 5 \frac{3}{4}}
$$

Solution:
1. Convert mixed numbers to improper fractions:
$$
2 \frac{1}{6} = \frac{13}{6}, \quad 4 \frac{2}{3} = \frac{14}{3}, \quad 2 \frac{1}{3} = \frac{7}{3}, \quad 5 \frac{3}{4} = \frac{23}{4}
$$
2. Simplify the numerator:
$$
\frac{13}{6} + \frac{14}{3}
$$
Find a common denominator (6):
$$
\frac{13}{6}, \quad \frac{14}{3} = \frac{28}{6}
$$
Add the fractions:
$$
\frac{13}{6} + \frac{28}{6} = \frac{41}{6}
$$
3. Simplify the denominator:
$$
\frac{7}{3} - \frac{23}{4}
$$
Find a common denominator (12):
$$
\frac{7}{3} = \frac{28}{12}, \quad \frac{23}{4} = \frac{69}{12}
$$
Subtract the fractions:
$$
\frac{28}{12} - \frac{69}{12} = \frac{-41}{12}
$$
4. Now divide the simplified numerator by the simplified denominator:
$$
\frac{\frac{41}{6}}{\frac{-41}{12}} = \frac{41}{6} \times \frac{12}{-41} = \frac{41 \times 12}{6 \times -41} = \frac{492}{-246} = -2
$$

Answer:
$$
\boxed{-2}
$$

---

#### Problem 82:
$$
\frac{1 \frac{1}{2} + 2 \frac{2}{3}}{1 \frac{1}{2} - 2 \frac{2}{3}}
$$

Solution:
1. Convert mixed numbers to improper fractions:
$$
1 \frac{1}{2} = \frac{3}{2}, \quad 2 \frac{2}{3} = \frac{8}{3}
$$
2. Simplify the numerator:
$$
\frac{3}{2} + \frac{8}{3}
$$
Find a common denominator (6):
$$
\frac{3}{2} = \frac{9}{6}, \quad \frac{8}{3} = \frac{16}{6}
$$
Add the fractions:
$$
\frac{9}{6} + \frac{16}{6} = \frac{25}{6}
$$
3. Simplify the denominator:
$$
\frac{3}{2} - \frac{8}{3}
$$
Find a common denominator (6):
$$
\frac{3}{2} = \frac{9}{6}, \quad \frac{8}{3} = \frac{16}{6}
$$
Subtract the fractions:
$$
\frac{9}{6} - \frac{16}{6} = \frac{-7}{6}
$$
4. Now divide the simplified numerator by the simplified denominator:
$$
\frac{\frac{25}{6}}{\frac{-7}{6}} = \frac{25}{6} \times \frac{6}{-7} = \frac{25 \times 6}{6 \times -7} = \frac{150}{-42} = \frac{-25}{7}
$$

Answer:
$$
\boxed{\frac{-25}{7}}
$$

---

Final Answers:

$$
\boxed{
\begin{aligned}
&67. \frac{2}{15}, \quad 68. \frac{32}{15}, \quad 69. \frac{25}{32}, \quad 70. 3, \\
&71. \frac{27}{8}, \quad 72. \frac{29}{6}, \quad 73. \frac{41}{10}, \quad 74. \frac{22}{7}, \\
&75. \frac{17}{44}, \quad 76. \frac{3}{5}, \quad 77. \frac{3}{7}, \quad 78. \frac{20}{3}, \\
&79. \frac{-14}{53}, \quad 80. \frac{-76}{129}, \quad 81. -2, \quad 82. \frac{-25}{7}.
\end{aligned}
}
$$