Worksheet for practicing division of fractions with whole numbers, themed as a math wizard challenge.

Problem Overview:

The task involves dividing mixed numbers by whole numbers. To solve these problems, we need to follow these steps:

1. Convert Mixed Numbers to Improper Fractions: A mixed number like \( 3 \frac{4}{8} \) can be converted to an improper fraction using the formula:
\[
\text{Improper Fraction} = \left( \text{Whole Number} \times \text{Denominator} \right) + \text{Numerator}
\]
The denominator remains the same.

2. Divide Fractions by Whole Numbers: Dividing a fraction by a whole number is equivalent to multiplying the fraction by the reciprocal of the whole number. For example:
\[
\frac{a}{b} \div c = \frac{a}{b} \times \frac{1}{c}
\]

3. Simplify the Result: After performing the division, simplify the resulting fraction if possible.

Let's solve each problem step by step.

---

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

#### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 8 \frac{1}{5} \):
\[
8 \frac{1}{5} = \frac{(8 \times 5) + 1}{5} = \frac{40 + 1}{5} = \frac{41}{5}
\]

- \( 3 \frac{4}{8} \):
\[
3 \frac{4}{8} = \frac{(3 \times 8) + 4}{8} = \frac{24 + 4}{8} = \frac{28}{8}
\]

#### Step 2: Simplify \( \frac{28}{8} \)
\[
\frac{28}{8} = \frac{7}{2}
\]

#### Step 3: Perform the Division
\[
\frac{41}{5} \div \frac{7}{2} = \frac{41}{5} \times \frac{2}{7} = \frac{41 \times 2}{5 \times 7} = \frac{82}{35}
\]

#### Step 4: Simplify the Result
\[
\frac{82}{35} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{\frac{82}{35}}
\]

---

Problem 2: \( 6 \frac{2}{3} \div 9 \frac{2}{3} \)

#### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 6 \frac{2}{3} \):
\[
6 \frac{2}{3} = \frac{(6 \times 3) + 2}{3} = \frac{18 + 2}{3} = \frac{20}{3}
\]

- \( 9 \frac{2}{3} \):
\[
9 \frac{2}{3} = \frac{(9 \times 3) + 2}{3} = \frac{27 + 2}{3} = \frac{29}{3}
\]

#### Step 2: Perform the Division
\[
\frac{20}{3} \div \frac{29}{3} = \frac{20}{3} \times \frac{3}{29} = \frac{20 \times 3}{3 \times 29} = \frac{60}{87}
\]

#### Step 3: Simplify the Result
\[
\frac{60}{87} = \frac{20}{29} \quad (\text{Divide numerator and denominator by } 3)
\]

Answer:
\[
\boxed{\frac{20}{29}}
\]

---

Problem 3: \( 3 \frac{1}{5} \div 5 \frac{2}{6} \)

#### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 3 \frac{1}{5} \):
\[
3 \frac{1}{5} = \frac{(3 \times 5) + 1}{5} = \frac{15 + 1}{5} = \frac{16}{5}
\]

- \( 5 \frac{2}{6} \):
\[
5 \frac{2}{6} = \frac{(5 \times 6) + 2}{6} = \frac{30 + 2}{6} = \frac{32}{6}
\]

#### Step 2: Simplify \( \frac{32}{6} \)
\[
\frac{32}{6} = \frac{16}{3}
\]

#### Step 3: Perform the Division
\[
\frac{16}{5} \div \frac{16}{3} = \frac{16}{5} \times \frac{3}{16} = \frac{16 \times 3}{5 \times 16} = \frac{48}{80}
\]

#### Step 4: Simplify the Result
\[
\frac{48}{80} = \frac{3}{5} \quad (\text{Divide numerator and denominator by } 16)
\]

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

---

Problem 4: \( 6 \frac{3}{4} \div 9 \frac{1}{2} \)

#### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 6 \frac{3}{4} \):
\[
6 \frac{3}{4} = \frac{(6 \times 4) + 3}{4} = \frac{24 + 3}{4} = \frac{27}{4}
\]

- \( 9 \frac{1}{2} \):
\[
9 \frac{1}{2} = \frac{(9 \times 2) + 1}{2} = \frac{18 + 1}{2} = \frac{19}{2}
\]

#### Step 2: Perform the Division
\[
\frac{27}{4} \div \frac{19}{2} = \frac{27}{4} \times \frac{2}{19} = \frac{27 \times 2}{4 \times 19} = \frac{54}{76}
\]

#### Step 3: Simplify the Result
\[
\frac{54}{76} = \frac{27}{38} \quad (\text{Divide numerator and denominator by } 2)
\]

Answer:
\[
\boxed{\frac{27}{38}}
\]

---

Problem 5: \( 1 \frac{1}{2} \div 8 \frac{2}{3} \)

#### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 1 \frac{1}{2} \):
\[
1 \frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}
\]

- \( 8 \frac{2}{3} \):
\[
8 \frac{2}{3} = \frac{(8 \times 3) + 2}{3} = \frac{24 + 2}{3} = \frac{26}{3}
\]

#### Step 2: Perform the Division
\[
\frac{3}{2} \div \frac{26}{3} = \frac{3}{2} \times \frac{3}{26} = \frac{3 \times 3}{2 \times 26} = \frac{9}{52}
\]

#### Step 3: Simplify the Result
\[
\frac{9}{52} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{\frac{9}{52}}
\]

---

Problem 6: \( 8 \frac{2}{3} \div 1 \frac{1}{2} \)

#### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 8 \frac{2}{3} \):
\[
8 \frac{2}{3} = \frac{(8 \times 3) + 2}{3} = \frac{24 + 2}{3} = \frac{26}{3}
\]

- \( 1 \frac{1}{2} \):
\[
1 \frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}
\]

#### Step 2: Perform the Division
\[
\frac{26}{3} \div \frac{3}{2} = \frac{26}{3} \times \frac{2}{3} = \frac{26 \times 2}{3 \times 3} = \frac{52}{9}
\]

#### Step 3: Simplify the Result
\[
\frac{52}{9} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{\frac{52}{9}}
\]

---

Problem 7: \( 5 \frac{2}{3} \div 9 \frac{2}{8} \)

#### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 5 \frac{2}{3} \):
\[
5 \frac{2}{3} = \frac{(5 \times 3) + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3}
\]

- \( 9 \frac{2}{8} \):
\[
9 \frac{2}{8} = \frac{(9 \times 8) + 2}{8} = \frac{72 + 2}{8} = \frac{74}{8}
\]

#### Step 2: Simplify \( \frac{74}{8} \)
\[
\frac{74}{8} = \frac{37}{4}
\]

#### Step 3: Perform the Division
\[
\frac{17}{3} \div \frac{37}{4} = \frac{17}{3} \times \frac{4}{37} = \frac{17 \times 4}{3 \times 37} = \frac{68}{111}
\]

#### Step 4: Simplify the Result
\[
\frac{68}{111} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{\frac{68}{111}}
\]

---

Problem 8: \( 7 \frac{1}{6} \div 1 \frac{3}{8} \)

#### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 7 \frac{1}{6} \):
\[
7 \frac{1}{6} = \frac{(7 \times 6) + 1}{6} = \frac{42 + 1}{6} = \frac{43}{6}
\]

- \( 1 \frac{3}{8} \):
\[
1 \frac{3}{8} = \frac{(1 \times 8) + 3}{8} = \frac{8 + 3}{8} = \frac{11}{8}
\]

#### Step 2: Perform the Division
\[
\frac{43}{6} \div \frac{11}{8} = \frac{43}{6} \times \frac{8}{11} = \frac{43 \times 8}{6 \times 11} = \frac{344}{66}
\]

#### Step 3: Simplify the Result
\[
\frac{344}{66} = \frac{172}{33} \quad (\text{Divide numerator and denominator by } 2)
\]

Answer:
\[
\boxed{\frac{172}{33}}
\]

---

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

#### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 9 \frac{2}{6} \):
\[
9 \frac{2}{6} = \frac{(9 \times 6) + 2}{6} = \frac{54 + 2}{6} = \frac{56}{6}
\]

#### Step 2: Simplify \( \frac{56}{6} \)
\[
\frac{56}{6} = \frac{28}{3}
\]

- \( 6 \frac{1}{2} \):
\[
6 \frac{1}{2} = \frac{(6 \times 2) + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2}
\]

#### Step 3: Perform the Division
\[
\frac{28}{3} \div \frac{13}{2} = \frac{28}{3} \times \frac{2}{13} = \frac{28 \times 2}{3 \times 13} = \frac{56}{39}
\]

#### Step 4: Simplify the Result
\[
\frac{56}{39} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{\frac{56}{39}}
\]

---

Problem 10: \( 7 \frac{1}{4} \div 9 \frac{4}{5} \)

#### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 7 \frac{1}{4} \):
\[
7 \frac{1}{4} = \frac{(7 \times 4) + 1}{4} = \frac{28 + 1}{4} = \frac{29}{4}
\]

- \( 9 \frac{4}{5} \):
\[
9 \frac{4}{5} = \frac{(9 \times 5) + 4}{5} = \frac{45 + 4}{5} = \frac{49}{5}
\]

#### Step 2: Perform the Division
\[
\frac{29}{4} \div \frac{49}{5} = \frac{29}{4} \times \frac{5}{49} = \frac{29 \times 5}{4 \times 49} = \frac{145}{196}
\]

#### Step 3: Simplify the Result
\[
\frac{145}{196} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{\frac{145}{196}}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ \frac{82}{35} \\
2. & \ \frac{20}{29} \\
3. & \ \frac{3}{5} \\
4. & \ \frac{27}{38} \\
5. & \ \frac{9}{52} \\
6. & \ \frac{52}{9} \\
7. & \ \frac{68}{111} \\
8. & \ \frac{172}{33} \\
9. & \ \frac{56}{39} \\
10. & \ \frac{145}{196}
\end{aligned}
}
\]