Here are the solutions to the fraction problems: 1) $(\frac{1}{2} \div \frac{1}{3}) \times \frac{1}{3}$ Step 1: Perform the division inside the parentheses. To divide by a fraction, multiply by its reciprocal. $$ \frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2} $$ Step 2: Multiply the result by $\frac{1}{3}$. $$ \frac{3}{2} \times \frac{1}{3} = \frac{3}{6} $$ Step 3: Simplify the fraction. $$ \frac{3}{6} = \frac{1}{2} $$ The final answer is $\boxed{\frac{1}{2}}$. 2) $\frac{1}{2} + \frac{2}{3} \times \frac{1}{4}$ Step 1: Perform the multiplication first. $$ \frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} $$ Simplify the fraction: $$ \frac{2}{12} = \frac{1}{6} $$ Step 2: Perform the addition. Find a common denominator for $\frac{1}{2}$ and $\frac{1}{6}$, which is 6. $$ \frac{1}{2} + \frac{1}{6} = \frac{1 \times 3}{2 \times 3} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{3+1}{6} = \frac{4}{6} $$ Step 3: Simplify the fraction. $$ \frac{4}{6} = \frac{2}{3} $$ The final answer is $\boxed{\frac{2}{3}}$. 3) $\frac{1}{4} - \frac{2}{3} + \frac{5}{6}$ Step 1: Find a common denominator for 4, 3, and 6, which is 12. Convert each fraction. $$ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} $$ $$ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} $$ $$ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} $$ Step 2: Perform the operations from left to right. $$ \frac{3}{12} - \frac{8}{12} + \frac{10}{12} = \frac{3-8}{12} + \frac{10}{12} = \frac{-5}{12} + \frac{10}{12} = \frac{-5+10}{12} = \frac{5}{12} $$ The final answer is $\boxed{\frac{5}{12}}$. 4) $\frac{3}{4} \div \frac{1}{2} \times \frac{1}{6}$ Step 1: Perform the division first (from left to right). To divide by a fraction, multiply by its reciprocal. $$ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} $$ Simplify the fraction: $$ \frac{6}{4} = \frac{3}{2} $$ Step 2: Multiply the result by $\frac{1}{6}$. $$ \frac{3}{2} \times \frac{1}{6} = \frac{3}{12} $$ Step 3: Simplify the fraction. $$ \frac{3}{12} = \frac{1}{4} $$ The final answer is $\boxed{\frac{1}{4}}$. 5) $\frac{2}{3} \text{ of } 2\frac{1}{4} \div \frac{1}{4}$ Step 1: Convert the mixed number to an improper fraction. $$ 2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4} $$ Step 2: Replace "of" with multiplication and perform the multiplication. $$ \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} $$ Simplify the fraction: $$ \frac{18}{12} = \frac{3}{2} $$ Step 3: Perform the division. To divide by a fraction, multiply by its reciprocal. $$ \frac{3}{2} \div \frac{1}{4} = \frac{3}{2} \times \frac{4}{1} = \frac{12}{2} $$ Step 4: Simplify the result. $$ \frac{12}{2} = 6 $$ The final answer is $\boxed{6}$. 6) $\frac{2}{3} - \frac{5}{6} + \frac{1}{2}$ Step 1: Find a common denominator for 3, 6, and 2, which is 6. Convert each fraction. $$ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} $$ $$ \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} $$ Step 2: Perform the operations from left to right. $$ \frac{4}{6} - \frac{5}{6} + \frac{3}{6} = \frac{4-5}{6} + \frac{3}{6} = \frac{-1}{6} + \frac{3}{6} = \frac{-1+3}{6} = \frac{2}{6} $$ Step 3: Simplify the fraction. $$ \frac{2}{6} = \frac{1}{3} $$ The final answer is $\boxed{\frac{1}{3}}$. 7) $\frac{1}{2} \div \frac{4}{3}$ Step 1: To divide by a fraction, multiply by its reciprocal. $$ \frac{1}{2} \div \frac{4}{3} = \frac{1}{2}