Convert the recurring decimal 0.\u0305\u030553 (with 53 repeating) into a fraction in lowest terms. Identify the correct equivalent fraction.

Introduction / Context:Recurring decimals can be expressed as exact fractions using a standard algebraic trick. Here, 0.\u0305\u030553 means the block “53” repeats indefinitely (0.535353…). The task is to convert this repeating decimal to a simplified fraction.

Given Data / Assumptions:

• Decimal: 0.535353… where the two-digit block 53 repeats.
• Let x represent the repeating decimal.

Concept / Approach:For a two-digit repetend, multiply x by 100 to align the repeating parts. Subtract the original x to eliminate the repeating tail, then solve for x as a ratio of integers. Reduce to lowest terms if needed.

Step-by-Step Solution:Let x = 0.535353…100x = 53.535353…Subtract: 100x − x = 53.535353… − 0.535353… = 53.Hence 99x = 53 ⇒ x = 53/99.

Verification / Alternative check:Divide 53 by 99 on a calculator to see the repeating pattern 0.535353…, confirming the exact match.

Why Other Options Are Wrong:

• 53/100 and 53/1000: These are terminating decimals, not repeating; they give 0.53 and 0.053 respectively.
• 53/90 and 26/45: Do not generate the exact 0.535353… sequence.

Common Pitfalls:Forgetting to multiply by the correct power of 10 (100 for a two-digit repetend), or reducing incorrectly.

Final Answer:53 / 99