Practice worksheet for converting improper fractions to mixed numbers and vice versa.

Problem: Converting Improper Fractions to Mixed Numbers and Mixed Numbers to Improper Fractions

The worksheet provided involves two main tasks:
1. Converting Improper Fractions to Mixed Numbers
2. Converting Mixed Numbers to Improper Fractions

Let's solve each part step by step.

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Part 1: Converting Improper Fractions to Mixed Numbers

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert an improper fraction to a mixed number:
1. Divide the numerator by the denominator.
2. The quotient becomes the whole number part.
3. The remainder becomes the numerator of the fractional part.
4. The denominator remains the same.

#### Solutions:

1. $\frac{29}{4}$
- Divide $29 \div 4$: Quotient = $7$, Remainder = $1$
- Mixed number: $7 \frac{1}{4}$
- Answer: $\boxed{7 \frac{1}{4}}$

2. $\frac{13}{6}$
- Divide $13 \div 6$: Quotient = $2$, Remainder = $1$
- Mixed number: $2 \frac{1}{6}$
- Answer: $\boxed{2 \frac{1}{6}}$

3. $\frac{73}{9}$
- Divide $73 \div 9$: Quotient = $8$, Remainder = $1$
- Mixed number: $8 \frac{1}{9}$
- Answer: $\boxed{8 \frac{1}{9}}$

4. $\frac{65}{8}$
- Divide $65 \div 8$: Quotient = $8$, Remainder = $1$
- Mixed number: $8 \frac{1}{8}$
- Answer: $\boxed{8 \frac{1}{8}}$

5. $\frac{17}{2}$
- Divide $17 \div 2$: Quotient = $8$, Remainder = $1$
- Mixed number: $8 \frac{1}{2}$
- Answer: $\boxed{8 \frac{1}{2}}$

6. $\frac{5}{2}$
- Divide $5 \div 2$: Quotient = $2$, Remainder = $1$
- Mixed number: $2 \frac{1}{2}$
- Answer: $\boxed{2 \frac{1}{2}}$

7. $\frac{25}{4}$
- Divide $25 \div 4$: Quotient = $6$, Remainder = $1$
- Mixed number: $6 \frac{1}{4}$
- Answer: $\boxed{6 \frac{1}{4}}$

8. $\frac{43}{7}$
- Divide $43 \div 7$: Quotient = $6$, Remainder = $1$
- Mixed number: $6 \frac{1}{7}$
- Answer: $\boxed{6 \frac{1}{7}}$

9. $\frac{29}{4}$
- Divide $29 \div 4$: Quotient = $7$, Remainder = $1$
- Mixed number: $7 \frac{1}{4}$
- Answer: $\boxed{7 \frac{1}{4}}$

10. $\frac{73}{9}$
- Divide $73 \div 9$: Quotient = $8$, Remainder = $1$
- Mixed number: $8 \frac{1}{9}$
- Answer: $\boxed{8 \frac{1}{9}}$

11. $\frac{19}{3}$
- Divide $19 \div 3$: Quotient = $6$, Remainder = $1$
- Mixed number: $6 \frac{1}{3}$
- Answer: $\boxed{6 \frac{1}{3}}$

12. $\frac{43}{7}$
- Divide $43 \div 7$: Quotient = $6$, Remainder = $1$
- Mixed number: $6 \frac{1}{7}$
- Answer: $\boxed{6 \frac{1}{7}}$

13. $\frac{11}{5}$
- Divide $11 \div 5$: Quotient = $2$, Remainder = $1$
- Mixed number: $2 \frac{1}{5}$
- Answer: $\boxed{2 \frac{1}{5}}$

14. $\frac{91}{10}$
- Divide $91 \div 10$: Quotient = $9$, Remainder = $1$
- Mixed number: $9 \frac{1}{10}$
- Answer: $\boxed{9 \frac{1}{10}}$

15. $\frac{37}{6}$
- Divide $37 \div 6$: Quotient = $6$, Remainder = $1$
- Mixed number: $6 \frac{1}{6}$
- Answer: $\boxed{6 \frac{1}{6}}$

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Part 2: Converting Mixed Numbers to Improper Fractions

A mixed number consists of a whole number and a proper fraction. To convert a mixed number to an improper fraction:
1. Multiply the whole number by the denominator of the fraction.
2. Add the result to the numerator of the fraction.
3. Write the sum as the new numerator, keeping the original denominator.

#### Solutions:

1. $7 \frac{1}{3}$
- Whole number = $7$, Denominator = $3$, Numerator = $1$
- New numerator = $(7 \times 3) + 1 = 21 + 1 = 22$
- Improper fraction: $\frac{22}{3}$
- Answer: $\boxed{\frac{22}{3}}$

2. $7 \frac{9}{10}$
- Whole number = $7$, Denominator = $10$, Numerator = $9$
- New numerator = $(7 \times 10) + 9 = 70 + 9 = 79$
- Improper fraction: $\frac{79}{10}$
- Answer: $\boxed{\frac{79}{10}}$

3. $7 \frac{3}{4}$
- Whole number = $7$, Denominator = $4$, Numerator = $3$
- New numerator = $(7 \times 4) + 3 = 28 + 3 = 31$
- Improper fraction: $\frac{31}{4}$
- Answer: $\boxed{\frac{31}{4}}$

4. $2 \frac{1}{2}$
- Whole number = $2$, Denominator = $2$, Numerator = $1$
- New numerator = $(2 \times 2) + 1 = 4 + 1 = 5$
- Improper fraction: $\frac{5}{2}$
- Answer: $\boxed{\frac{5}{2}}$

5. $8 \frac{4}{7}$
- Whole number = $8$, Denominator = $7$, Numerator = $4$
- New numerator = $(8 \times 7) + 4 = 56 + 4 = 60$
- Improper fraction: $\frac{60}{7}$
- Answer: $\boxed{\frac{60}{7}}$

6. $2 \frac{3}{5}$
- Whole number = $2$, Denominator = $5$, Numerator = $3$
- New numerator = $(2 \times 5) + 3 = 10 + 3 = 13$
- Improper fraction: $\frac{13}{5}$
- Answer: $\boxed{\frac{13}{5}}$

7. $3 \frac{5}{8}$
- Whole number = $3$, Denominator = $8$, Numerator = $5$
- New numerator = $(3 \times 8) + 5 = 24 + 5 = 29$
- Improper fraction: $\frac{29}{8}$
- Answer: $\boxed{\frac{29}{8}}$

8. $6 \frac{7}{9}$
- Whole number = $6$, Denominator = $9$, Numerator = $7$
- New numerator = $(6 \times 9) + 7 = 54 + 7 = 61$
- Improper fraction: $\frac{61}{9}$
- Answer: $\boxed{\frac{61}{9}}$

9. $9 \frac{1}{8}$
- Whole number = $9$, Denominator = $8$, Numerator = $1$
- New numerator = $(9 \times 8) + 1 = 72 + 1 = 73$
- Improper fraction: $\frac{73}{8}$
- Answer: $\boxed{\frac{73}{8}}$

10. $6 \frac{2}{5}$
- Whole number = $6$, Denominator = $5$, Numerator = $2$
- New numerator = $(6 \times 5) + 2 = 30 + 2 = 32$
- Improper fraction: $\frac{32}{5}$
- Answer: $\boxed{\frac{32}{5}}$

11. $4 \frac{1}{3}$
- Whole number = $4$, Denominator = $3$, Numerator = $1$
- New numerator = $(4 \times 3) + 1 = 12 + 1 = 13$
- Improper fraction: $\frac{13}{3}$
- Answer: $\boxed{\frac{13}{3}}$

12. $2 \frac{2}{3}$
- Whole number = $2$, Denominator = $3$, Numerator = $2$
- New numerator = $(2 \times 3) + 2 = 6 + 2 = 8$
- Improper fraction: $\frac{8}{3}$
- Answer: $\boxed{\frac{8}{3}}$

13. $8 \frac{1}{2}$
- Whole number = $8$, Denominator = $2$, Numerator = $1$
- New numerator = $(8 \times 2) + 1 = 16 + 1 = 17$
- Improper fraction: $\frac{17}{2}$
- Answer: $\boxed{\frac{17}{2}}$

14. $4 \frac{3}{10}$
- Whole number = $4$, Denominator = $10$, Numerator = $3$
- New numerator = $(4 \times 10) + 3 = 40 + 3 = 43$
- Improper fraction: $\frac{43}{10}$
- Answer: $\boxed{\frac{43}{10}}$

15. $8 \frac{4}{5}$
- Whole number = $8$, Denominator = $5$, Numerator = $4$
- New numerator = $(8 \times 5) + 4 = 40 + 4 = 44$
- Improper fraction: $\frac{44}{5}$
- Answer: $\boxed{\frac{44}{5}}$

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Final Answers:

#### Part 1: Converting Improper Fractions to Mixed Numbers
1. $\boxed{7 \frac{1}{4}}$
2. $\boxed{2 \frac{1}{6}}$
3. $\boxed{8 \frac{1}{9}}$
4. $\boxed{8 \frac{1}{8}}$
5. $\boxed{8 \frac{1}{2}}$
6. $\boxed{2 \frac{1}{2}}$
7. $\boxed{6 \frac{1}{4}}$
8. $\boxed{6 \frac{1}{7}}$
9. $\boxed{7 \frac{1}{4}}$
10. $\boxed{8 \frac{1}{9}}$
11. $\boxed{6 \frac{1}{3}}$
12. $\boxed{6 \frac{1}{7}}$
13. $\boxed{2 \frac{1}{5}}$
14. $\boxed{9 \frac{1}{10}}$
15. $\boxed{6 \frac{1}{6}}$

#### Part 2: Converting Mixed Numbers to Improper Fractions
1. $\boxed{\frac{22}{3}}$
2. $\boxed{\frac{79}{10}}$
3. $\boxed{\frac{31}{4}}$
4. $\boxed{\frac{5}{2}}$
5. $\boxed{\frac{60}{7}}$
6. $\boxed{\frac{13}{5}}$
7. $\boxed{\frac{29}{8}}$
8. $\boxed{\frac{61}{9}}$
9. $\boxed{\frac{73}{8}}$
10. $\boxed{\frac{32}{5}}$
11. $\boxed{\frac{13}{3}}$
12. $\boxed{\frac{8}{3}}$
13. $\boxed{\frac{17}{2}}$
14. $\boxed{\frac{43}{10}}$
15. $\boxed{\frac{44}{5}}$

Final Answer:
\[
\boxed{
\begin{array}{l}
\text{Part 1: } 7 \frac{1}{4}, 2 \frac{1}{6}, 8 \frac{1}{9}, 8 \frac{1}{8}, 8 \frac{1}{2}, 2 \frac{1}{2}, 6 \frac{1}{4}, 6 \frac{1}{7}, 7 \frac{1}{4}, 8 \frac{1}{9}, 6 \frac{1}{3}, 6 \frac{1}{7}, 2 \frac{1}{5}, 9 \frac{1}{10}, 6 \frac{1}{6} \\
\text{Part 2: } \frac{22}{3}, \frac{79}{10}, \frac{31}{4}, \frac{5}{2}, \frac{60}{7}, \frac{13}{5}, \frac{29}{8}, \frac{61}{9}, \frac{73}{8}, \frac{32}{5}, \frac{13}{3}, \frac{8}{3}, \frac{17}{2}, \frac{43}{10}, \frac{44}{5}
\end{array}
}
\]