Convert 11.42424242... To a rational expression in the form of a/b, where b ≠ 0

Answer 1

To convert the repeating decimal \( 11.42424242\ldots \) into a rational expression, we can follow these steps:

1. Let \( x \) equal the repeating decimal: \[ x = 11.42424242\ldots \]

2. Isolate the repeating part:
   The decimal has the integer part (11) and the repeating part (0.42424242...). We can separate the integer part: \[ x = 11 + 0.42424242\ldots \]

3. Let \( y \) represent the repeating part: \[ y = 0.42424242\ldots \]

4. Set up an equation for \( y \):
   Notice that the repeating part "42" has 2 digits: \[ 100y = 42.42424242\ldots \]

5. Subtract the original \( y \) from this equation: \[ 100y - y = 42.42424242\ldots - 0.42424242\ldots \] \[ 99y = 42 \]

6. Solve for \( y \): \[ y = \frac{42}{99} \]

7. Simplify the fraction \( \frac{42}{99} \): The greatest common divisor (GCD) of 42 and 99 is 3: \[ y = \frac{42 \div 3}{99 \div 3} = \frac{14}{33} \]

8. Combine with the integer part to find \( x \): \[ x = 11 + y = 11 + \frac{14}{33} \]

9. Convert the integer into a fraction with the same denominator: \[ x = \frac{11 \cdot 33}{33} + \frac{14}{33} = \frac{363}{33} + \frac{14}{33} = \frac{363 + 14}{33} = \frac{377}{33} \]

10. Final expression:
    Therefore, the repeating decimal \( 11.42424242\ldots \) as a rational expression is: \[ \boxed{\frac{377}{33}} \]