What is 1.21 repeating as a fraction?
The correct answer and explanation is:
To convert the repeating decimal 1.2121‾1.21\overline{21} into a fraction, follow these steps:
- Represent the number algebraically: Let x=1.2121‾x = 1.21\overline{21}. This means that the digits “21” repeat indefinitely after the decimal point.
- Remove the repeating decimal: To eliminate the repeating part, multiply both sides of the equation by 100. This gives: 100x=121.2121‾100x = 121.21\overline{21} Notice that the decimal part of both xx and 100x100x is the same (i.e., 0.21\overline{21}).
- Subtract the original equation from this new equation: Subtract the equation x=1.2121‾x = 1.21\overline{21} from 100x=121.2121‾100x = 121.21\overline{21}: 100x−x=121.2121‾−1.2121‾100x – x = 121.21\overline{21} – 1.21\overline{21} This simplifies to: 99x=12099x = 120
- Solve for xx: Now divide both sides of the equation by 99 to isolate xx: x=12099x = \frac{120}{99}
- Simplify the fraction: To simplify the fraction, find the greatest common divisor (GCD) of 120 and 99. The GCD is 3. Divide both the numerator and denominator by 3: x=120÷399÷3=4033x = \frac{120 \div 3}{99 \div 3} = \frac{40}{33}
Thus, 1.2121‾1.21\overline{21} as a fraction is 4033\frac{40}{33}.
This process works because multiplying by powers of 10 shifts the decimal point, and subtracting removes the repeating part, leaving a solvable equation.