9. Evaluate each of the following: \( \begin{array}{llll}\text { (i) } 2 \frac{1}{2}+3 \frac{1}{4} & \text { (ii) } 8 \frac{1}{5}+2 \frac{1}{3} & \text { (iii) } 5 \frac{1}{3}-1 \frac{1}{4} & \text { (iv) } 10 \frac{1}{12}-3 \frac{3}{4}\end{array} \)

To evaluate each of the expressions, we will convert the mixed numbers to improper fractions, perform the operations, and then convert back to mixed numbers if necessary. ### (i) \( 2 \frac{1}{2} + 3 \frac{1}{4} \) 1. Convert to improper fractions: \[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2} \] \[ 3 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{13}{4} \] 2. Find a common denominator (which is 4): \[ \frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4} \] 3. Add the fractions: \[ \frac{10}{4} + \frac{13}{4} = \frac{10 + 13}{4} = \frac{23}{4} \] 4. Convert back to a mixed number: \[ \frac{23}{4} = 5 \frac{3}{4} \] ### Result for (i): \[ 2 \frac{1}{2} + 3 \frac{1}{4} = 5 \frac{3}{4} \] --- ### (ii) \( 8 \frac{1}{5} + 2 \frac{1}{3} \) 1. Convert to improper fractions: \[ 8 \frac{1}{5} = \frac{8 \times 5 + 1}{5} = \frac{41}{5} \] \[ 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3} \] 2. Find a common denominator (which is 15): \[ \frac{41}{5} = \frac{41 \times 3}{5 \times 3} = \frac{123}{15} \] \[ \frac{7}{3} = \frac{7 \times 5}{3 \times 5} = \frac{35}{15} \] 3. Add the fractions: \[ \frac{123}{15} + \frac{35}{15} = \frac{123 + 35}{15} = \frac{158}{15} \] 4. Convert back to a mixed number: \[ \frac{158}{15} = 10 \frac{8}{15} \] ### Result for (ii): \[ 8 \frac{1}{5} + 2 \frac{1}{3} = 10 \frac{8}{15} \] --- ### (iii) \( 5 \frac{1}{3} - 1 \frac{1}{4} \) 1. Convert to improper fractions: \[ 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{16}{3} \] \[ 1 \frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{5}{4} \] 2. Find a common denominator (which is 12): \[ \frac{16}{3} = \frac{16 \times 4}{3 \times 4} = \frac{64}{12} \] \[ \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12} \] 3. Subtract the fractions: \[ \frac{64}{12} - \frac{15}{12} = \frac{64 - 15}{12} = \frac{49}{12} \] 4. Convert back to a mixed number: \[ \frac{49}{12} = 4 \frac{1}{12} \] ### Result for (iii): \[ 5 \frac{1}{3} - 1 \frac{1}{4} = 4 \frac{1}{12} \] --- ### (iv) \( 10 \frac{1}{12} - 3 \frac{3}{4} \) 1. Convert to improper fractions: \[ 10 \frac{1}{12} = \frac{10 \times 12 + 1}{12} = \frac{121}{12} \] \[ 3 \frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{15}{4} \] 2. Find a common denominator (which is 12): \[ \frac{15}{4} = \frac{15 \times 3}{4 \times 3} = \frac{45}{12} \] 3. Subtract the fractions: \[ \frac{121}{12} - \frac{45}{12} = \frac{121 - 45}{12} = \frac{76}{12} \] 4. Simplify: \[ \frac{76}{12} = \frac{19}{3} \] 5. Convert back to a mixed number: \[ \frac{19}{3} = 6 \frac{1}{3} \] ### Result for (iv): \[ 10 \frac{1}{12} - 3 \frac{3}{4} = 6 \frac{1}{3} \] --- ### Final Results: \[ \begin{array}{llll} \text{(i)} & 5 \frac{3}{4} \\ \text{(ii)} & 10 \frac{8}{15} \\ \text{(iii)} & 4 \frac{1}{12} \\ \text{(iv)} & 6 \frac{1}{3} \\ \end{array} \]