Let \( x = 11.42424242 \ldots \).
We can separate the non-repeating and repeating parts of the decimal: \[ x = 11.4 + 0.02424242 \ldots \] Let \( y = 0.02424242 \ldots \).
To convert \( y \) into a fraction, we can represent it as follows: \[ y = 0.02424242 \ldots \] Multiplying both sides by 100 (to shift the decimal point two places to the right): \[ 100y = 2.42424242 \ldots \] Now, notice that \( 2.42424242 \ldots \) can be rewritten by separating the whole number part: \[ 100y = 2 + 0.42424242 \ldots \]
Let \( z = 0.42424242 \ldots \). We can express \( z \) similarly: \[ z = 0.42424242 \ldots \] Multiplying both sides by 100: \[ 100z = 42.42424242 \ldots \] Rearranging gives: \[ 100z = 42 + z \] Thus: \[ 100z - z = 42 \] \[ 99z = 42 \] \[ z = \frac{42}{99} \] Now, we simplify \( z \): \[ z = \frac{14}{33} \quad \text{(dividing the numerator and denominator by 3)} \]
Next, substitute \( z \) back into our equation for \( y \): \[ 100y = 2 + \frac{14}{33} \] Convert \( 2 \) to a fraction with a denominator of 33: \[ 2 = \frac{66}{33} \] So, \[ 100y = \frac{66}{33} + \frac{14}{33} = \frac{80}{33} \] Thus, \[ y = \frac{80}{3300} = \frac{8}{330} = \frac{4}{165} \quad \text{(dividing by 2)} \]
Now, substitute \( y \) back to the original equation: \[ x = 11.4 + y = 11.4 + \frac{4}{165} \] Convert \( 11.4 \) to a fraction: \[ 11.4 = \frac{114}{10} = \frac{57}{5} \] To add \( \frac{57}{5} \) and \( \frac{4}{165} \), we need a common denominator. The least common multiple of 5 and 165 is 165.
Convert \( \frac{57}{5} \): \[ \frac{57}{5} = \frac{57 \times 33}{5 \times 33} = \frac{1881}{165} \]
Now, add the fractions: \[ x = \frac{1881}{165} + \frac{4}{165} = \frac{1881 + 4}{165} = \frac{1885}{165} \]
The final answer is: \[ \frac{1885}{165} \]
This fraction can be simplified further: \[ \frac{1885 \div 5}{165 \div 5} = \frac{377}{33} \]
Thus, the rational expression representing \( 11.42424242 \ldots \) is: \[ \frac{377}{33} \]