\( ( 3 , \overline { 2 } + 5 , \overline { 7 } ) \times 0 , \overline { 3 } + 25 \cdot ( 0 , \overline { 6 } - 0,6 ) \cdot \frac { 0.5 } { 0.5 } \)

To solve the expression \( ( 3 , \overline { 2 } + 5 , \overline { 7 } ) \times 0 , \overline { 3 } + 25 \cdot ( 0 , \overline { 6 } - 0,6 ) \cdot \frac { 0.5 } { 0.5 } \), we need to break it down step by step. 1. **Understanding the notation**: - The notation \( 0, \overline{3} \) represents the repeating decimal \( 0.3333\ldots \), which is equal to \( \frac{1}{3} \). - Similarly, \( 0, \overline{6} \) represents \( 0.6666\ldots \), which is equal to \( \frac{2}{3} \). - The notation \( 3, \overline{2} \) represents \( 3.2222\ldots \), which is equal to \( 3 + \frac{2}{9} = \frac{29}{9} \). - The notation \( 5, \overline{7} \) represents \( 5.7777\ldots \), which is equal to \( 5 + \frac{7}{90} = \frac{517}{90} \). 2. **Calculating the components**: - First, we calculate \( 0, \overline{6} - 0,6 \): \[ 0, \overline{6} = \frac{2}{3}, \quad 0.6 = \frac{3}{5} \] To subtract these, we need a common denominator: \[ \frac{2}{3} - \frac{3}{5} = \frac{10}{15} - \frac{9}{15} = \frac{1}{15} \] 3. **Calculating the multiplication**: - Now we calculate \( 25 \cdot (0, \overline{6} - 0.6) \): \[ 25 \cdot \frac{1}{15} = \frac{25}{15} = \frac{5}{3} \] 4. **Calculating the entire expression**: - Now we can substitute back into the original expression: \[ (3, \overline{2} + 5, \overline{7}) \times 0, \overline{3} + \frac{5}{3} \] - We need to calculate \( 3, \overline{2} + 5, \overline{7} \): \[ 3, \overline{2} = \frac{29}{9}, \quad 5, \overline{7} = \frac{517}{90} \] To add these, we need a common denominator: \[ \frac{29}{9} = \frac{290}{90}, \quad \frac{517}{90} \] Thus, \[ \frac{290}{90} + \frac{517}{90} = \frac{807}{90} \] 5. **Final calculation**: - Now we have: \[ \frac{807}{90} \times 0, \overline{3} + \frac{5}{3} \] where \( 0, \overline{3} = \frac{1}{3} \): \[ \frac{807}{90} \times \frac{1}{3} = \frac{807}{270} \] - Now we add \( \frac{5}{3} \): \[ \frac{5}{3} = \frac{50}{30} = \frac{1350}{270} \] Thus, \[ \frac{807}{270} + \frac{1350}{270} = \frac{2157}{270} \] 6. **Final result**: - The final result is: \[ \frac{2157}{270} \] This fraction can be simplified if needed, but it is the final answer in its current form.