1. **State the problem:** Simplify expressions containing mixed numbers by converting to improper fractions, then perform addition, subtraction, multiplication, or division, and simplify. 2. **Convert mixed numbers to improper fractions:** - a) $3\frac{1}{2} = \frac{7}{2},\quad 4\frac{1}{3} = \frac{13}{3},\quad 5\frac{1}{4} = \frac{21}{4}$ - b) $6\frac{1}{2} = \frac{13}{2},\quad 1\frac{5}{8} = \frac{13}{8},\quad 2\frac{1}{3} = \frac{7}{3}$ - c) $1\frac{2}{3} = \frac{5}{3},\quad 1\frac{3}{4} = \frac{7}{4},\quad 2\frac{5}{8} = \frac{21}{8}$ - d) $3\frac{1}{3} = \frac{10}{3},\quad 1\frac{1}{6} = \frac{7}{6}$ 3. **Solve each part:** **a)** Sum $\frac{7}{2} + \frac{13}{3} + \frac{21}{4}$. - Common denominator is 12. - Convert fractions: $\frac{7}{2} = \frac{42}{12},\quad \frac{13}{3} = \frac{52}{12},\quad \frac{21}{4} = \frac{63}{12}$ - Sum: $\frac{42}{12} + \frac{52}{12} + \frac{63}{12} = \frac{157}{12}$ - Convert improper fraction to mixed: $\frac{157}{12} = 13\frac{1}{12}$ **b)** Compute $(1\frac{5}{8} + 2\frac{1}{3})$ and subtract from $6\frac{1}{2}$. - Add inside parentheses: - Convert to improper fractions: $\frac{13}{8} + \frac{7}{3}$ - Common denominator 24. - Convert: $\frac{13}{8} = \frac{39}{24},\quad \frac{7}{3} = \frac{56}{24}$ - Sum: $\frac{39}{24} + \frac{56}{24} = \frac{95}{24}$ - Convert $6\frac{1}{2} = \frac{13}{2} = \frac{156}{24}$ - Subtract: $\frac{156}{24} - \frac{95}{24} = \frac{61}{24}$ - Convert to mixed: $2\frac{13}{24}$ **c)** Divide inside parentheses and subtract from $1\frac{2}{3}$. - Divide $1\frac{3}{4} \div 2\frac{5}{8}$: - Convert: $\frac{7}{4} \div \frac{21}{8} = \frac{7}{4} \times \frac{8}{21} = \frac{56}{84} = \frac{2}{3}$ - Subtract from $1\frac{2}{3} = \frac{5}{3}$: - $\frac{5}{3} - \frac{2}{3} = \frac{3}{3} = 1$ **d)** Add inside parentheses and multiply by 2. - Add: $3\frac{1}{3} + 1\frac{1}{6} = \frac{10}{3} + \frac{7}{6}$ - Common denominator 6. - Convert: $\frac{10}{3} = \frac{20}{6}$ - Sum: $\frac{20}{6} + \frac{7}{6} = \frac{27}{6} = \frac{9}{2}$ - Multiply by 2: $2 \times \frac{9}{2} = 9$ **Final answers:** - a) $13\frac{1}{12}$ - b) $2\frac{13}{24}$ - c) $1$ - d) $9$