To express \( 3.011111\overline{1} \) (where the digit "1" repeats indefinitely) as a mixed number, follow these steps:

1. Let \( x = 3.011111\overline{1} \).

2. Subtract 3 from both sides to isolate the repeating decimal part:
   \[
   x - 3 = 0.011111\overline{1}.
   \]

3. Now, let \( y = 0.011111\overline{1} \).

4. Multiply \( y \) by 10 to shift the repeating decimal:
   \[
   10y = 0.111111\overline{1}.
   \]

5. Subtract \( y \) from \( 10y \):
   \[
   10y - y = 0.111111\overline{1} - 0.011111\overline{1} = 0.1.
   \]

   So, \( 9y = 0.1 \).

6. Solve for \( y \):
   \[
   y = \frac{0.1}{9} = \frac{1}{90}.
   \]

7. Now, go back to the original equation \( x = 3 + y \):
   \[
   x = 3 + \frac{1}{90}.
   \]

Thus, \( 3.011111\overline{1} \) as a mixed number is \( 3 \frac{1}{90} \).

Would you like more details on any part of the explanation?

Here are 5 related questions:
1. How do you convert repeating decimals into fractions in general?
2. How do you express a terminating decimal as a fraction?
3. Can a repeating decimal be converted into an improper fraction?
4. How would you simplify a fraction like \( \frac{1}{90} \)?
5. What are the steps for converting a non-repeating decimal into a fraction?

**Tip:** When converting repeating decimals to fractions, isolating the repeating part is key for a smooth calculation!

Learn how to express 3.011111 (with repeating 1) as a mixed number. This detailed solution walks through isolating the repeating decimal and converting it into a fraction, resulting in the mixed number 3 1/90. Suitable for Grades 8-10, this example reinforces key algebraic concepts like working with fractions and repeating decimals.

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