1. **Problem Statement:** Convert improper fractions to mixed numbers and mixed numbers to improper fractions. 2. **Formulas and Rules:** - To convert an improper fraction $\frac{a}{b}$ to a mixed number, divide $a$ by $b$ to get quotient $q$ and remainder $r$. The mixed number is $q \frac{r}{b}$. - To convert a mixed number $q \frac{r}{b}$ to an improper fraction, use $\frac{bq + r}{b}$. 3. **Convert improper fractions to mixed numbers:** (a) $\frac{7}{4}$: $7 \div 4 = 1$ remainder $3$, so $1 \frac{3}{4}$. (b) $\frac{13}{6}$: $13 \div 6 = 2$ remainder $1$, so $2 \frac{1}{6}$. (c) $\frac{19}{5}$: $19 \div 5 = 3$ remainder $4$, so $3 \frac{4}{5}$. (d) $\frac{32}{7}$: $32 \div 7 = 4$ remainder $4$, so $4 \frac{4}{7}$. (e) $\frac{43}{8}$: $43 \div 8 = 5$ remainder $3$, so $5 \frac{3}{8}$. (f) $\frac{67}{10}$: $67 \div 10 = 6$ remainder $7$, so $6 \frac{7}{10}$. 4. **Convert mixed numbers to improper fractions:** (a) $2 \frac{3}{4}$: $2 \times 4 + 3 = 8 + 3 = 11$, so $\frac{11}{4}$. (b) $3 \frac{2}{7}$: $3 \times 7 + 2 = 21 + 2 = 23$, so $\frac{23}{7}$. (c) $4 \frac{5}{8}$: $4 \times 8 + 5 = 32 + 5 = 37$, so $\frac{37}{8}$. (d) $7 \frac{4}{9}$: $7 \times 9 + 4 = 63 + 4 = 67$, so $\frac{67}{9}$. (e) $10 \frac{3}{8}$: $10 \times 8 + 3 = 80 + 3 = 83$, so $\frac{83}{8}$. (f) $6 \frac{9}{10}$: $6 \times 10 + 9 = 60 + 9 = 69$, so $\frac{69}{10}$. **Final answers:** Improper to mixed: (a) $1 \frac{3}{4}$, (b) $2 \frac{1}{6}$, (c) $3 \frac{4}{5}$, (d) $4 \frac{4}{7}$, (e) $5 \frac{3}{8}$, (f) $6 \frac{7}{10}$. Mixed to improper: (a) $\frac{11}{4}$, (b) $\frac{23}{7}$, (c) $\frac{37}{8}$, (d) $\frac{67}{9}$, (e) $\frac{83}{8}$, (f) $\frac{69}{10}$.