1. **Stating the problem:** Calculate the value of the expression $$3,6 \cdot 0,1\overline{1} + 2,1\overline{1} : 0,83\overline{3} - 0,93\overline{3}$$ where the overline denotes repeating decimals. 2. **Convert repeating decimals to fractions:** - $0,1\overline{1} = 0.111\ldots = \frac{1}{9}$ - $2,1\overline{1} = 2 + 0,1\overline{1} = 2 + \frac{1}{9} = \frac{19}{9}$ - $0,83\overline{3} = 0.8333\ldots = \frac{5}{6}$ - $0,93\overline{3} = 0.9333\ldots = \frac{14}{15}$ 3. **Rewrite the expression with fractions:** $$3.6 \cdot \frac{1}{9} + \frac{19}{9} : \frac{5}{6} - \frac{14}{15}$$ 4. **Convert 3.6 to fraction:** $$3.6 = \frac{36}{10} = \frac{18}{5}$$ 5. **Calculate each part:** - Multiply: $$\frac{18}{5} \cdot \frac{1}{9} = \frac{18 \cdot 1}{5 \cdot 9} = \frac{18}{45}$$ - Simplify with cancellation: $$\frac{\cancel{18}^{2 \cdot 9}}{\cancel{45}^{5 \cdot 9}} = \frac{2}{5}$$ 6. **Division:** $$\frac{19}{9} : \frac{5}{6} = \frac{19}{9} \cdot \frac{6}{5} = \frac{19 \cdot 6}{9 \cdot 5} = \frac{114}{45}$$ - Simplify with cancellation: $$\frac{\cancel{114}^{6 \cdot 19}}{\cancel{45}^{9 \cdot 5}} = \frac{38}{15}$$ 7. **Now the expression is:** $$\frac{2}{5} + \frac{38}{15} - \frac{14}{15}$$ 8. **Find common denominator 15:** $$\frac{2}{5} = \frac{6}{15}$$ 9. **Sum and subtract:** $$\frac{6}{15} + \frac{38}{15} - \frac{14}{15} = \frac{6 + 38 - 14}{15} = \frac{30}{15} = 2$$ **Final answer:** $$2$$