When the recurring decimal 0.232323... is converted into an equivalent fraction in simplest form, what is the resulting fraction?

Introduction / Context:
Repeating or recurring decimals can always be expressed as rational numbers, that is, as fractions of integers. This question asks you to convert the recurring decimal 0.232323... (with 23 repeating) into a fraction in simplest form. Understanding this conversion is important for aptitude tests and basic algebra.

Given Data / Assumptions:

• The decimal is 0.232323..., where the block "23" repeats indefinitely.
• We must express this as a fraction in lowest terms.
• No rounding is involved because the pattern is perfectly repeating.

Concept / Approach:
For a recurring decimal with two repeating digits, like 0.abab..., we can use a standard algebraic trick. Let x be the repeating decimal. Multiplying x by 100 shifts the decimal two places to the right. Subtracting x then cancels the repeating part, leaving a simple linear equation that we can solve for x as a fraction.

Step-by-Step Solution:
Step 1: Let x = 0.232323... Step 2: Multiply both sides by 100 to shift two decimal places: 100x = 23.232323... Step 3: Subtract the original x: 100x - x = 23.232323... - 0.232323... Step 4: The repeating parts cancel, giving 99x = 23. Step 5: Solve for x: x = 23 / 99. Step 6: Check for further simplification. Since 23 is a prime number and does not divide 99, the fraction is already in simplest form.

Verification / Alternative check:
Compute 23 / 99 as a decimal by long division. You will get 0.232323..., confirming that we have the correct fraction. The repeating block "23" appears continuously, matching the original decimal representation.

Why Other Options Are Wrong:
1/5 = 0.2, which does not repeat as 0.232323....
2/9 ≈ 0.222..., repeating only the digit 2, not "23".
23/100 = 0.23, which terminates and does not continue repeating.
3/10 = 0.3, again a terminating decimal, not a recurring one with block "23".

Common Pitfalls:
Learners sometimes confuse terminating and recurring decimals or incorrectly multiply by 10 instead of 100 when two digits repeat. It is crucial to match the number of zeros in the multiplier to the number of digits in the repeating block. Also, always check whether the fraction can be simplified further by checking common factors.

Final Answer:
The recurring decimal 0.232323... is equal to the fraction 23/99 in simplest form.