Convert a recurring decimal to a fraction: Express the repeating decimal 4.12 (where 12 repeats) as a single simplified fraction.

Introduction / Context:
Recurring decimals such as 4.121212... can be converted exactly to rational numbers. Mastering this technique helps move between decimal and fractional forms without rounding.

Given Data / Assumptions:

• Repeating decimal: 4.12 with the block 12 repeating indefinitely (4.121212...).
• Find an exact fraction in simplest terms.

Concept / Approach:
Let x = 4.121212.... For a two-digit repeat, multiply by 100 to shift the decimal so that subtraction eliminates the repeating part.

Step-by-Step Solution:

Verification / Alternative check:
Write the integer part plus the repeating fractional part: 4 + 0.121212...; and 0.121212... = 12/99 = 4/33. Then 4 + 4/33 = (132 + 4)/33 = 136/33, confirming the result.

Why Other Options Are Wrong:
411/99 corresponds to 4.1515..., not 4.1212.... 52/9 equals approximately 5.777..., too large. 411/90 ≈ 4.566..., also incorrect.

Common Pitfalls:
Using 10x instead of 100x for a two-digit repeat, or failing to simplify the resulting fraction completely.

Final Answer:
136/33