Grade 5 worksheet on dividing mixed numbers by mixed numbers, designed for K-5 students.

Problem: Dividing Mixed Numbers by Mixed Numbers

The task is to solve the division of mixed numbers for each problem listed in the worksheet. Here's how to approach each problem step by step.

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#### Step-by-Step Solution:

1. Convert Mixed Numbers to Improper Fractions:
- A mixed number $ a \frac{b}{c} $ can be converted to an improper fraction using the formula:
$$
a \frac{b}{c} = \frac{(a \times c) + b}{c}
$$

2. Divide Fractions:
- To divide two fractions, multiply the first fraction by the reciprocal of the second fraction:
$$
\frac{p}{q} \div \frac{r}{s} = \frac{p}{q} \times \frac{s}{r} = \frac{p \times s}{q \times r}
$$

3. Simplify the Result:
- Simplify the resulting fraction if possible.
- Convert the result back to a mixed number if it is an improper fraction.

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Solutions to Each Problem:

#### Problem 1: $ 9 \frac{5}{6} \div 1 \frac{1}{3} $

1. Convert mixed numbers to improper fractions:
- $ 9 \frac{5}{6} = \frac{(9 \times 6) + 5}{6} = \frac{54 + 5}{6} = \frac{59}{6} $
- $ 1 \frac{1}{3} = \frac{(1 \times 3) + 1}{3} = \frac{3 + 1}{3} = \frac{4}{3} $

2. Divide the fractions:
$$
\frac{59}{6} \div \frac{4}{3} = \frac{59}{6} \times \frac{3}{4} = \frac{59 \times 3}{6 \times 4} = \frac{177}{24}
$$

3. Simplify the result:
- Simplify $ \frac{177}{24} $ by dividing both numerator and denominator by their greatest common divisor (GCD), which is 3:
$$
\frac{177 \div 3}{24 \div 3} = \frac{59}{8}
$$

4. Convert to a mixed number:
- $ \frac{59}{8} = 7 \frac{3}{8} $ (since $ 59 \div 8 = 7 $ remainder $ 3 $)

Answer:
$$
\boxed{7 \frac{3}{8}}
$$

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#### Problem 2: $ 6 \frac{10}{13} \div 4 \frac{2}{5} $

1. Convert mixed numbers to improper fractions:
- $ 6 \frac{10}{13} = \frac{(6 \times 13) + 10}{13} = \frac{78 + 10}{13} = \frac{88}{13} $
- $ 4 \frac{2}{5} = \frac{(4 \times 5) + 2}{5} = \frac{20 + 2}{5} = \frac{22}{5} $

2. Divide the fractions:
$$
\frac{88}{13} \div \frac{22}{5} = \frac{88}{13} \times \frac{5}{22} = \frac{88 \times 5}{13 \times 22} = \frac{440}{286}
$$

3. Simplify the result:
- Simplify $ \frac{440}{286} $ by dividing both numerator and denominator by their GCD, which is 22:
$$
\frac{440 \div 22}{286 \div 22} = \frac{20}{13}
$$

4. Convert to a mixed number:
- $ \frac{20}{13} = 1 \frac{7}{13} $ (since $ 20 \div 13 = 1 $ remainder $ 7 $)

Answer:
$$
\boxed{1 \frac{7}{13}}
$$

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#### Problem 3: $ 3 \frac{1}{6} \div 2 \frac{3}{4} $

1. Convert mixed numbers to improper fractions:
- $ 3 \frac{1}{6} = \frac{(3 \times 6) + 1}{6} = \frac{18 + 1}{6} = \frac{19}{6} $
- $ 2 \frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4} $

2. Divide the fractions:
$$
\frac{19}{6} \div \frac{11}{4} = \frac{19}{6} \times \frac{4}{11} = \frac{19 \times 4}{6 \times 11} = \frac{76}{66}
$$

3. Simplify the result:
- Simplify $ \frac{76}{66} $ by dividing both numerator and denominator by their GCD, which is 2:
$$
\frac{76 \div 2}{66 \div 2} = \frac{38}{33}
$$

4. Convert to a mixed number:
- $ \frac{38}{33} = 1 \frac{5}{33} $ (since $ 38 \div 33 = 1 $ remainder $ 5 $)

Answer:
$$
\boxed{1 \frac{5}{33}}
$$

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#### Problem 4: $ 4 \frac{2}{7} \div 2 \frac{5}{7} $

1. Convert mixed numbers to improper fractions:
- $ 4 \frac{2}{7} = \frac{(4 \times 7) + 2}{7} = \frac{28 + 2}{7} = \frac{30}{7} $
- $ 2 \frac{5}{7} = \frac{(2 \times 7) + 5}{7} = \frac{14 + 5}{7} = \frac{19}{7} $

2. Divide the fractions:
$$
\frac{30}{7} \div \frac{19}{7} = \frac{30}{7} \times \frac{7}{19} = \frac{30 \times 7}{7 \times 19} = \frac{210}{133}
$$

3. Simplify the result:
- Simplify $ \frac{210}{133} $ by dividing both numerator and denominator by their GCD, which is 7:
$$
\frac{210 \div 7}{133 \div 7} = \frac{30}{19}
$$

4. Convert to a mixed number:
- $ \frac{30}{19} = 1 \frac{11}{19} $ (since $ 30 \div 19 = 1 $ remainder $ 11 $)

Answer:
$$
\boxed{1 \frac{11}{19}}
$$

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#### Problem 5: $ 9 \frac{2}{3} \div 5 \frac{2}{3} $

1. Convert mixed numbers to improper fractions:
- $ 9 \frac{2}{3} = \frac{(9 \times 3) + 2}{3} = \frac{27 + 2}{3} = \frac{29}{3} $
- $ 5 \frac{2}{3} = \frac{(5 \times 3) + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3} $

2. Divide the fractions:
$$
\frac{29}{3} \div \frac{17}{3} = \frac{29}{3} \times \frac{3}{17} = \frac{29 \times 3}{3 \times 17} = \frac{87}{51}
$$

3. Simplify the result:
- Simplify $ \frac{87}{51} $ by dividing both numerator and denominator by their GCD, which is 3:
$$
\frac{87 \div 3}{51 \div 3} = \frac{29}{17}
$$

4. Convert to a mixed number:
- $ \frac{29}{17} = 1 \frac{12}{17} $ (since $ 29 \div 17 = 1 $ remainder $ 12 $)

Answer:
$$
\boxed{1 \frac{12}{17}}
$$

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#### Problem 6: $ 1 \frac{3}{4} \div 4 \frac{7}{8} $

1. Convert mixed numbers to improper fractions:
- $ 1 \frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4} $
- $ 4 \frac{7}{8} = \frac{(4 \times 8) + 7}{8} = \frac{32 + 7}{8} = \frac{39}{8} $

2. Divide the fractions:
$$
\frac{7}{4} \div \frac{39}{8} = \frac{7}{4} \times \frac{8}{39} = \frac{7 \times 8}{4 \times 39} = \frac{56}{156}
$$

3. Simplify the result:
- Simplify $ \frac{56}{156} $ by dividing both numerator and denominator by their GCD, which is 4:
$$
\frac{56 \div 4}{156 \div 4} = \frac{14}{39}
$$

Answer:
$$
\boxed{\frac{14}{39}}
$$

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#### Problem 7: $ 8 \frac{5}{6} \div 2 \frac{2}{3} $

1. Convert mixed numbers to improper fractions:
- $ 8 \frac{5}{6} = \frac{(8 \times 6) + 5}{6} = \frac{48 + 5}{6} = \frac{53}{6} $
- $ 2 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} $

2. Divide the fractions:
$$
\frac{53}{6} \div \frac{8}{3} = \frac{53}{6} \times \frac{3}{8} = \frac{53 \times 3}{6 \times 8} = \frac{159}{48}
$$

3. Simplify the result:
- Simplify $ \frac{159}{48} $ by dividing both numerator and denominator by their GCD, which is 3:
$$
\frac{159 \div 3}{48 \div 3} = \frac{53}{16}
$$

4. Convert to a mixed number:
- $ \frac{53}{16} = 3 \frac{5}{16} $ (since $ 53 \div 16 = 3 $ remainder $ 5 $)

Answer:
$$
\boxed{3 \frac{5}{16}}
$$

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#### Problem 8: $ 8 \frac{1}{12} \div 3 \frac{3}{4} $

1. Convert mixed numbers to improper fractions:
- $ 8 \frac{1}{12} = \frac{(8 \times 12) + 1}{12} = \frac{96 + 1}{12} = \frac{97}{12} $
- $ 3 \frac{3}{4} = \frac{(3 \times 4) + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4} $

2. Divide the fractions:
$$
\frac{97}{12} \div \frac{15}{4} = \frac{97}{12} \times \frac{4}{15} = \frac{97 \times 4}{12 \times 15} = \frac{388}{180}
$$

3. Simplify the result:
- Simplify $ \frac{388}{180} $ by dividing both numerator and denominator by their GCD, which is 4:
$$
\frac{388 \div 4}{180 \div 4} = \frac{97}{45}
$$

4. Convert to a mixed number:
- $ \frac{97}{45} = 2 \frac{7}{45} $ (since $ 97 \div 45 = 2 $ remainder $ 7 $)

Answer:
$$
\boxed{2 \frac{7}{45}}
$$

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#### Problem 9: $ 6 \frac{3}{5} \div 3 \frac{1}{2} $

1. Convert mixed numbers to improper fractions:
- $ 6 \frac{3}{5} = \frac{(6 \times 5) + 3}{5} = \frac{30 + 3}{5} = \frac{33}{5} $
- $ 3 \frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} $

2. Divide the fractions:
$$
\frac{33}{5} \div \frac{7}{2} = \frac{33}{5} \times \frac{2}{7} = \frac{33 \times 2}{5 \times 7} = \frac{66}{35}
$$

3. Simplify the result:
- $ \frac{66}{35} $ is already in simplest form.

4. Convert to a mixed number:
- $ \frac{66}{35} = 1 \frac{31}{35} $ (since $ 66 \div 35 = 1 $ remainder $ 31 $)

Answer:
$$
\boxed{1 \frac{31}{35}}
$$

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#### Problem 10: $ 5 \frac{2}{9} \div 4 \frac{1}{3} $

1. Convert mixed numbers to improper fractions:
- $ 5 \frac{2}{9} = \frac{(5 \times 9) + 2}{9} = \frac{45 + 2}{9} = \frac{47}{9} $
- $ 4 \frac{1}{3} = \frac{(4 \times 3) + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3} $

2. Divide the fractions:
$$
\frac{47}{9} \div \frac{13}{3} = \frac{47}{9} \times \frac{3}{13} = \frac{47 \times 3}{9 \times 13} = \frac{141}{117}
$$

3. Simplify the result:
- Simplify $ \frac{141}{117} $ by dividing both numerator and denominator by their GCD, which is 3:
$$
\frac{141 \div 3}{117 \div 3} = \frac{47}{39}
$$

4. Convert to a mixed number:
- $ \frac{47}{39} = 1 \frac{8}{39} $ (since $ 47 \div 39 = 1 $ remainder $ 8 $)

Answer:
$$
\boxed{1 \frac{8}{39}}
$$

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Final Answers:

1. $ \boxed{7 \frac{3}{8}} $
2. $ \boxed{1 \frac{7}{13}} $
3. $ \boxed{1 \frac{5}{33}} $
4. $ \boxed{1 \frac{11}{19}} $
5. $ \boxed{1 \frac{12}{17}} $
6. $ \boxed{\frac{14}{39}} $
7. $ \boxed{3 \frac{5}{16}} $
8. $ \boxed{2 \frac{7}{45}} $
9. $ \boxed{1 \frac{31}{35}} $
10. $ \boxed{1 \frac{8}{39}} $

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This completes the solution for all problems. If you have any further questions, feel free to ask!