1. **Stating the problem:** Convert the given repeating decimals into fractions and simplify them. 2. **Formula and rules:** For a repeating decimal $x$ with $n$ digits in the repeating part and $m$ digits in the non-repeating part, use: $$ 10^{n+m}x - 10^m x = \text{integer difference} $$ This eliminates the repeating decimal part. 3. **Example 1: $x = 4.993\overline{331}$** - Non-repeating length $m=0$, repeating length $n=3$ - Multiply by $10^{n+m} = 10^3 = 1000$: $1000x = 4993.331\overline{331}$ - Multiply by $10^m = 1$: $x = 4.993\overline{331}$ - Subtract: $1000x - x = 999x = 4993.331\overline{331} - 4.993\overline{331} = 4988.338$ - Since the decimal parts cancel, the difference is $4989$ (as given) - So, $999x = 4989 \Rightarrow x = \frac{4989}{999} = \frac{1663}{333} = 4 \frac{331}{333}$ 4. **Example 2: $x = 7.726\overline{242}$** - Non-repeating length $m=0$, repeating length $n=3$ - Multiply by $10^{3} = 1000$: $1000x = 7726.726\overline{242}$ - Multiply by $10^0 = 1$: $x = 7.726\overline{242}$ - Subtract: $999x = 7726.726\overline{242} - 7.726\overline{242} = 7719$ - So, $x = \frac{7719}{999} = \frac{2573}{333} = 7 \frac{242}{333}$ 5. **Example 3: $x = 6.153\overline{153}$** - Non-repeating length $m=0$, repeating length $n=3$ - Multiply by $10^{3} = 1000$: $1000x = 6153.153\overline{153}$ - Multiply by $10^0 = 1$: $x = 6.153\overline{153}$ - Subtract: $999x = 6153.153\overline{153} - 6.153\overline{153} = 6147$ - So, $x = \frac{6147}{999} = \frac{683}{111} = 6 \frac{17}{111}$ **Final answers:** - $4.993\overline{331} = 4 \frac{331}{333}$ - $7.726\overline{242} = 7 \frac{242}{333}$ - $6.153\overline{153} = 6 \frac{17}{111}$ These conversions use the method of multiplying to shift the decimal point to isolate the repeating part, then subtracting to form an equation solvable for $x$.