Compute. \( \frac{5}{34} \cdot 22 \frac{2}{3}-2 \frac{1}{9} \)

To compute the expression \( \frac{5}{34} \cdot 22 \frac{2}{3}-2 \frac{1}{9} \), we first convert the mixed numbers into improper fractions. 1. Convert \( 22 \frac{2}{3} \) to an improper fraction: \[ 22 \frac{2}{3} = 22 \cdot 3 + 2 = 66 + 2 = 68 \quad \Rightarrow \quad \frac{68}{3} \] 2. Convert \( 2 \frac{1}{9} \) to an improper fraction: \[ 2 \frac{1}{9} = 2 \cdot 9 + 1 = 18 + 1 = 19 \quad \Rightarrow \quad \frac{19}{9} \] Now we can substitute these values back into the expression: \[ \frac{5}{34} \cdot \frac{68}{3} - \frac{19}{9} \] Next, we need to perform the multiplication: \[ \frac{5 \cdot 68}{34 \cdot 3} = \frac{340}{102} \] Now simplify \( \frac{340}{102} \): \[ \text{The GCD of 340 and 102 is 34.} \] \[ \frac{340 \div 34}{102 \div 34} = \frac{10}{3} \] Now we have: \[ \frac{10}{3} - \frac{19}{9} \] To subtract these fractions, we need a common denominator. The least common multiple of 3 and 9 is 9. So we convert \( \frac{10}{3} \): \[ \frac{10}{3} = \frac{10 \cdot 3}{3 \cdot 3} = \frac{30}{9} \] Now we can subtract: \[ \frac{30}{9} - \frac{19}{9} = \frac{30 - 19}{9} = \frac{11}{9} \] Thus, the final result is: \[ \frac{11}{9} \] If you need to present this as a mixed number, it would be: \[ 1 \frac{2}{9} \] So, \( \frac{5}{34} \cdot 22 \frac{2}{3} - 2 \frac{1}{9} = \frac{11}{9} \) or \( 1 \frac{2}{9}. \)