1. Problem: Simplify the following expressions: a) $3 \frac{1}{2} + 4 \frac{1}{3} + 5 \frac{1}{4}$ b) $6 \frac{1}{2} - (1 \frac{5}{8} + 2 \frac{1}{3})$ c) $1 \frac{2}{3} - \left(1 \frac{3}{4} \div 2 \frac{5}{8}\right)$ d) $2 \times \left(3 \frac{1}{3} + 1 \frac{1}{6}\right)$ 2. Convert all mixed numbers to improper fractions: a) $3 \frac{1}{2} = \frac{7}{2}$, $4 \frac{1}{3} = \frac{13}{3}$, $5 \frac{1}{4} = \frac{21}{4}$ b) $6 \frac{1}{2} = \frac{13}{2}$, $1 \frac{5}{8} = \frac{13}{8}$, $2 \frac{1}{3} = \frac{7}{3}$ c) $1 \frac{2}{3} = \frac{5}{3}$, $1 \frac{3}{4} = \frac{7}{4}$, $2 \frac{5}{8} = \frac{21}{8}$ d) $3 \frac{1}{3} = \frac{10}{3}$, $1 \frac{1}{6} = \frac{7}{6}$ 3. Solve each part: a) Add $\frac{7}{2} + \frac{13}{3} + \frac{21}{4}$ Find common denominator: 12 Convert: $\frac{7}{2} = \frac{42}{12}$, $\frac{13}{3} = \frac{52}{12}$, $\frac{21}{4} = \frac{63}{12}$ Sum: $\frac{42}{12} + \frac{52}{12} + \frac{63}{12} = \frac{157}{12}$ Convert to mixed number: $12$ goes into $157$ thirteen times with remainder $1$, so $13 \frac{1}{12}$ b) Compute inside the parentheses first: $1 \frac{5}{8} + 2 \frac{1}{3} = \frac{13}{8} + \frac{7}{3}$ Find common denominator: 24 Convert: $\frac{13}{8} = \frac{39}{24}$, $\frac{7}{3} = \frac{56}{24}$ Sum: $\frac{39}{24} + \frac{56}{24} = \frac{95}{24}$ Now subtract from $6 \frac{1}{2} = \frac{13}{2}$: Convert $\frac{13}{2}$ to denominator 24: $\frac{156}{24}$ Subtract: $\frac{156}{24} - \frac{95}{24} = \frac{61}{24}$ Convert to mixed number: $2 \frac{13}{24}$ c) Divide inside the parentheses: $1 \frac{3}{4} \div 2 \frac{5}{8} = \frac{7}{4} \div \frac{21}{8} = \frac{7}{4} \times \frac{8}{21} = \frac{56}{84} = \frac{2}{3}$ Subtract from $1 \frac{2}{3} = \frac{5}{3}$: $\frac{5}{3} - \frac{2}{3} = \frac{3}{3} = 1$ d) Add inside the parentheses: $3 \frac{1}{3} + 1 \frac{1}{6} = \frac{10}{3} + \frac{7}{6}$ Find 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) $9$