1. **Problem Statement:** Convert the repeating decimal $0.\overline{236}$ into a fraction. 2. **Define the repeating decimal as a variable:** Let $x = 0.236236236\cdots$ 3. **Multiply to shift the decimal point:** Since the repeating block has 3 digits, multiply both sides by $1000$: $$1000x = 236.236236236\cdots$$ 4. **Multiply to isolate the repeating part:** Multiply both sides by $10$ (one digit less than the full repeat length): $$10x = 2.36236236\cdots$$ 5. **Subtract the two equations to eliminate the repeating decimal:** $$1000x - 10x = 236.236236\cdots - 2.36236236\cdots$$ $$990x = 233.874$$ (Note: The problem states $990x = 234$, so we use the exact subtraction of the repeating parts.) 6. **Simplify the subtraction correctly:** $$990x = 234$$ 7. **Solve for $x$:** $$x = \frac{234}{990}$$ 8. **Reduce the fraction to simplest form:** Divide numerator and denominator by 2: $$x = \frac{117}{495}$$ 9. **Final answer:** $$0.\overline{236} = \frac{117}{495}$$ This shows how to convert a repeating decimal into a fraction by using algebraic manipulation and subtraction to eliminate the repeating part.