The repeating decimal $0.24(2)$ represents the number $0.2422222...$, where the digit 2 repeats indefinitely. To convert this repeating decimal to a fraction, follow these steps:

Step 1: Define the repeating decimal

Let: $x = 0.2422222\ldots$ This means that $x = 0.24\overline{2}$, where the bar indicates that the digit 2 repeats.

Step 2: Eliminate the repeating part

To deal with the repeating decimal, multiply both sides of the equation by 10, to shift the decimal point one place to the right: $10x = 2.4222222\ldots$

Now subtract the original equation from this new equation to eliminate the repeating part.

$10x - x = 2.4222222\ldots - 0.2422222\ldots$ This simplifies to: $9x = 2.18$

Step 3: Solve for $x$

Now, divide both sides of the equation by 9 to solve for $x$: $x = \frac{2.18}{9}$

Step 4: Simplify the fraction

To remove the decimal in the numerator, multiply both the numerator and the denominator by 100: $x = \frac{218}{900}$

Step 5: Simplify the fraction further

Now, simplify the fraction by finding the greatest common divisor (GCD) of 218 and 900. The GCD is 2, so: $x = \frac{218 \div 2}{900 \div 2} = \frac{109}{450}$

Thus, the fraction representation of $0.24\overline{2}$ is: $x = \frac{109}{450}$

Final Answer:

The generating fraction of $0.24\overline{2}$ is $\frac{109}{450}$.

Would you like more details or have any questions about the solution? Here are some related questions to expand your understanding:

1. How do you convert a decimal like $0.5\overline{3}$ into a fraction?
2. What is the process for converting a non-repeating decimal to a fraction?
3. How can you express a repeating decimal with multiple repeating digits (e.g., $0.\overline{123}$) as a fraction?
4. How does simplifying fractions help in understanding the relation between a repeating decimal and its fractional form?
5. What is the general method for converting any repeating decimal to a fraction?

Tip: When dealing with repeating decimals, one common strategy is to set up an equation where the variable represents the repeating decimal, then shift the decimal point to align the repeating parts for subtraction.