When the repeating decimal 0.090909... is written as a fraction in simplest form, what is that fraction?

Introduction / Context:
This question tests your understanding of how to convert a repeating decimal into an exact fraction. Repeating decimals often appear in algebra and number system problems, and knowing how to handle them shows deeper conceptual understanding.

Given Data / Assumptions:

• Repeating decimal: 0.090909..., where 09 repeats indefinitely.
• We must express this as a fraction in simplest form.
• The dots indicate that the pattern 09 continues without end.

Concept / Approach:
To convert a repeating decimal into a fraction, we use an algebraic method. We denote the decimal by a variable, multiply by a suitable power of 10 so that the repeating part lines up, and then subtract to eliminate the repetition.

Step-by-Step Solution:

Verification / Alternative check:
We know that 1/11 is 0.090909..., so converting back to a decimal immediately confirms that our fraction form is correct.

Why Other Options Are Wrong:
1/33 equals approximately 0.030303..., which has a different repeating block. 2/33 equals about 0.060606..., also incorrect. 6/11 equals about 0.545454..., which is much larger and contains 54 as the repeating pair, not 09.

Common Pitfalls:
A common error is misidentifying the length of the repeating block or multiplying by 10 instead of 100 in this case. Another mistake is simplifying 9/99 incorrectly or not simplifying it at all, forgetting to reduce the fraction to lowest terms.

Final Answer:
The repeating decimal 0.090909... is equal to the fraction 1/11.