Prove algebraically that 1.018 (with 18 recurring) = 56/55

To prove that a recurring decimal is equal to a certain fraction, we must convert the decimal into a fraction algebraically. 1. Let `x` be the recurring decimal: `x = 1.0181818...` 2. We want to create two equations where the part after the decimal point is the same, so we can subtract them and eliminate the recurring part. The recurring part is `18`. 3. Multiply `x` by 10 to move the decimal point just before the recurring part begins: `10x = 10.181818...` (Equation 1) 4. Multiply `x` by 1000 to move the decimal point after the first full cycle of the recurring part: `1000x = 1018.181818...` (Equation 2) 5. Now, subtract Equation 1 from Equation 2: `1000x - 10x = 1018.181818... - 10.181818...` `990x = 1008` 6. Solve for `x` by dividing by 990: `x = 1008 / 990` 7. Simplify the fraction. We can divide both the numerator and denominator by their common factors. - Divide by 2: `x = 504 / 495` - The sum of the digits of 504 is 9, and the sum of the digits of 495 is 18. Both are divisible by 9. - 504 ÷ 9 = 56 - 495 ÷ 9 = 55 - So, `x = 56/55`. We have shown that 1.018 (recurring) is equal to 56/55.