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

Introduction / Context:
This question examines your understanding of recurring decimals and how to convert them into rational numbers in fractional form. Knowing how to express repeating decimals as fractions is important in number system topics and helps connect decimal representation with exact fractional values used in algebra and higher mathematics.

Given Data / Assumptions:

• The given decimal is 0.232323..., where the block 23 repeats indefinitely.
• We assume the repetition continues forever without stopping.
• We must find a fraction in lowest terms that is exactly equal to this recurring decimal.
• No approximation or rounding is allowed; the answer must be exact.

Concept / Approach:
The standard method for converting a recurring decimal into a fraction uses algebraic manipulation. We assign the repeating decimal to a variable, multiply by a power of 10 that shifts the repeating block to the left of the decimal point, and then subtract the original expression. This cancellation removes the repeating part and yields a simple linear equation in one variable.

Step-by-Step Solution:
Let x = 0.232323... where 23 repeats indefinitely.Since two digits repeat, multiply x by 100: 100x = 23.232323...Now subtract the original x from this: 100x - x = 23.232323... - 0.232323...This gives 99x = 23.Solve for x: x = 23 / 99.Check if 23 / 99 is already in simplest form. The numerator 23 is a prime number and does not divide 99, so the fraction is already reduced.Therefore, 0.232323... = 23 / 99 in simplest fractional form.

Verification / Alternative check:
We can verify by performing long division of 23 by 99. The quotient begins 0.232323..., and the block 23 will repeat indefinitely. This confirms that 23 / 99 is indeed the exact representation of the recurring decimal. Using a calculator in repeating mode or manual division both support this conclusion and show that our algebraic method is reliable.

Why Other Options Are Wrong:
Option 1/5 equals 0.2, which does not have a repeating block 23. Option 2/9 equals 0.222..., so the repeating block is 2, not 23. Option 23/100 equals 0.23 exactly, with no repetition, so it is a terminating decimal. Option 21/90 simplifies to 7/30, which equals 0.2333..., again not matching 0.232323... . None of these capture the specific two digit repeating pattern that appears in the given decimal.

Common Pitfalls:
Students sometimes multiply by 10 instead of 100, which fails to align the repeating block properly. Another mistake is to forget to subtract the original x, leaving the repeating part in place. Some learners also mistakenly round 0.232323... to 0.23, treating it as a terminating decimal, which leads to 23/100 and an incorrect answer. Careful attention to the length of the repeating block and disciplined use of algebra avoid these issues.

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