1. **Problem Statement:** Rewrite the repeating decimal $1.83\overline{3}$ as a simplified fraction. 2. **Understanding the repeating decimal:** The number $1.83\overline{3}$ means $1.8333\ldots$ where the digit 3 repeats indefinitely. 3. **Set up an equation:** Let $x = 1.8333\ldots$ 4. **Multiply to isolate the repeating part:** Since only one digit (3) repeats, multiply by 10 to shift one decimal place: $$10x = 18.3333\ldots$$ 5. **Subtract original equation from this:** $$10x - x = 18.3333\ldots - 1.8333\ldots$$ $$9x = 16.5$$ 6. **Solve for $x$:** $$x = \frac{16.5}{9}$$ 7. **Convert decimal numerator to fraction:** $$16.5 = \frac{33}{2}$$ 8. **Rewrite $x$ as:** $$x = \frac{33/2}{9} = \frac{33}{2} \times \frac{1}{9} = \frac{33}{18}$$ 9. **Simplify the fraction:** $$\frac{33}{18} = \frac{11}{6}$$ 10. **Final answer:** $$1.83\overline{3} = \frac{11}{6}$$ This fraction is in simplest form because 11 and 6 share no common factors other than 1.