When the recurring decimal 0.090909... is converted into a fraction in lowest terms, what is the resulting fraction?

Introduction / Context:
Many competitive exams include questions on converting recurring (repeating) decimals into fractions. A recurring decimal has one or more digits that repeat in a fixed pattern. In this question, 0.090909... has the repeating block 09. We must convert this recurring decimal into a rational number in lowest terms.

Given Data / Assumptions:

• Recurring decimal: 0.090909..., where the repeating block is 09.
• The repetition continues indefinitely.
• We must express this number as a fraction in simplest form.

Concept / Approach:
For a recurring decimal with a repeating block, we use the standard algebraic technique:

• Let x be the recurring decimal.
• Multiply x by a power of 10 that shifts one full repeat block to the left of the decimal point.
• Subtract the original x from this new expression to eliminate the repeating part.
• Solve for x as a fraction and simplify if needed.

Step-by-Step Solution:
Step 1: Let x = 0.090909... . Step 2: The repeating block is two digits (09), so multiply both sides by 100: 100x = 9.090909... . Step 3: Note that 100x and x share the same repeating part .090909..., which will cancel when we subtract. Step 4: Subtract the original equation from the new one: 100x - x = 9.090909... - 0.090909.... Step 5: This gives 99x = 9. Step 6: Solve for x: x = 9 / 99. Step 7: Simplify 9/99 by dividing numerator and denominator by 9: 9/99 = 1/11.

Verification / Alternative Check:
We know that 1/11 equals 0.090909... by standard fraction-decimal conversions. Dividing 1 by 11 produces 0.09 with 09 repeating forever. This confirms that our algebraic result is correct.

Why Other Options Are Wrong:
1/33 gives a smaller repeating decimal approximately equal to 0.030303..., not 0.090909....
2/33 corresponds to about 0.060606..., which still has a different value.
6/11 is approximately 0.545454..., whose repeating block is 54, not 09. Therefore, these options do not match the given decimal.

Common Pitfalls:
A common error is to incorrectly choose the power of 10, especially when the repeating block has two or more digits. Another mistake is simplifying the resulting fraction incorrectly. Always factor both numerator and denominator to remove any common divisor completely so that the answer is in lowest terms.

Final Answer:
The recurring decimal 0.090909... is equal to the fraction 1/11 in simplest form.