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To express the repeating decimal \(0.\overline{418}\) as a fraction, we'll follow the steps shown in the image, using algebra to solve for \(x\).

1. Let \(x = 0.\overline{418}\).

2. Multiply both sides by 1000 (since there are three digits in the repeating sequence) to shift the decimal point three places to the right:
   \[
   1000x = 418.181818\ldots
   \]

3. Now, subtract the original equation from this new equation:
   \[
   1000x - x = 418.181818\ldots - 0.418181818\ldots
   \]
   Simplifying the right-hand side, we remove the repeating decimals:
   \[
   999x = 418 - 0.418181818\ldots
   \]
   \[
   999x = 417.581818\ldots
   \]

4. Now, solving for \(x\):
   \[
   x = \frac{417.581818\ldots}{999}
   \]
   However, we recognize the repeating decimal is exactly \(\frac{418}{1000}\) less, so simplifying,
   \[
   x = \frac{418}{1000} - \frac{1}{1000}
   \]
   \[
   x = \frac{417}{999}
   \]

Thus, the repeating decimal \(0.\overline{418}\) as a fraction is \(\frac{417}{999}\).

Would you like any details on these steps or do you have any further questions? Here are some follow-up queries you might find useful:

1. How do you determine the number to multiply \(x\) by for different lengths of repeating decimals?
2. What is the process for simplifying the fraction \(\frac{417}{999}\)?
3. Can the same method be used for decimals that have non-repeating parts before the repeating sequence begins?
4. How do you verify the accuracy of the fraction obtained from a repeating decimal?
5. Are there any tools or calculators that can automate this process?

**Tip:** When dealing with repeating decimals, always ensure to multiply by a power of 10 that matches the number of repeating digits to shift the decimal point past the entire repeating sequence.

Express 0.418 (with repeating 18) as a fraction. Show all your working.

Learn how to express the repeating decimal 0.418181818... as a fraction using algebra. This step-by-step solution covers key concepts in algebra and fractions, perfect for students in Grades 8-10.

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