1. **State the problem:** We need to add and subtract mixed numbers and fractions in each expression. 2. **Convert mixed numbers to improper fractions:** - For example, $4\ \frac{2}{3} = \frac{14}{3}$ because $4 = \frac{12}{3}$ and $12+2=14$. 3. **Find a common denominator:** - To add or subtract fractions, convert all fractions to have the same denominator. - The least common denominator (LCD) for denominators 2, 3, 4, 5, 6, 8, 9, 10, 12 is 120 or smaller depending on the problem. 4. **Rewrite each fraction with the LCD and perform the operations:** --- **Problem 1:** $\frac{1}{2} + 4\ \frac{2}{3} - 3\ \frac{3}{4}$ - Convert mixed numbers: $4\ \frac{2}{3} = \frac{14}{3}$ $3\ \frac{3}{4} = \frac{15}{4}$ - LCD of denominators 2, 3, 4 is 12. - Convert each fraction: $\frac{1}{2} = \frac{6}{12}$ $\frac{14}{3} = \frac{56}{12}$ $\frac{15}{4} = \frac{45}{12}$ - Perform addition and subtraction: $$\frac{6}{12} + \frac{56}{12} - \frac{45}{12} = \frac{6 + 56 - 45}{12} = \frac{17}{12}$$ - Simplify if needed: $\frac{17}{12}$ is an improper fraction, can be written as $1\ \frac{5}{12}$. --- **Problem 2:** $\frac{2}{9} + 3\ \frac{5}{8} - \frac{1}{3}$ - Convert mixed number: $3\ \frac{5}{8} = \frac{29}{8}$ - LCD of denominators 9, 8, 3 is 72. - Convert each fraction: $\frac{2}{9} = \frac{16}{72}$ $\frac{29}{8} = \frac{261}{72}$ $\frac{1}{3} = \frac{24}{72}$ - Perform operations: $$\frac{16}{72} + \frac{261}{72} - \frac{24}{72} = \frac{16 + 261 - 24}{72} = \frac{253}{72}$$ - Simplify: $\frac{253}{72} = 3\ \frac{37}{72}$. --- **Problem 3:** $\frac{7}{9} + 1\ \frac{1}{3} - \frac{3}{5}$ - Convert mixed number: $1\ \frac{1}{3} = \frac{4}{3}$ - LCD of denominators 9, 3, 5 is 45. - Convert each fraction: $\frac{7}{9} = \frac{35}{45}$ $\frac{4}{3} = \frac{60}{45}$ $\frac{3}{5} = \frac{27}{45}$ - Perform operations: $$\frac{35}{45} + \frac{60}{45} - \frac{27}{45} = \frac{35 + 60 - 27}{45} = \frac{68}{45}$$ - Simplify: $\frac{68}{45} = 1\ \frac{23}{45}$. --- **Problem 4:** $3\ \frac{1}{2} + \frac{7}{10} - \frac{3}{5}$ - Convert mixed number: $3\ \frac{1}{2} = \frac{7}{2}$ - LCD of denominators 2, 10, 5 is 10. - Convert each fraction: $\frac{7}{2} = \frac{35}{10}$ $\frac{7}{10} = \frac{7}{10}$ $\frac{3}{5} = \frac{6}{10}$ - Perform operations: $$\frac{35}{10} + \frac{7}{10} - \frac{6}{10} = \frac{35 + 7 - 6}{10} = \frac{36}{10}$$ - Simplify: $\frac{36}{10} = 3\ \frac{3}{5}$. --- **Problem 5:** $\frac{8}{9} + \frac{3}{4} - \frac{1}{6}$ - LCD of denominators 9, 4, 6 is 36. - Convert each fraction: $\frac{8}{9} = \frac{32}{36}$ $\frac{3}{4} = \frac{27}{36}$ $\frac{1}{6} = \frac{6}{36}$ - Perform operations: $$\frac{32}{36} + \frac{27}{36} - \frac{6}{36} = \frac{32 + 27 - 6}{36} = \frac{53}{36}$$ - Simplify: $\frac{53}{36} = 1\ \frac{17}{36}$. --- **Final answers:** 1. $1\ \frac{5}{12}$ 2. $3\ \frac{37}{72}$ 3. $1\ \frac{23}{45}$ 4. $3\ \frac{3}{5}$ 5. $1\ \frac{17}{36}$