What is 0.12 repeating expressed as a fraction in simplest form

What is 0.12 repeating expressed as a fraction in simplest form

The Correct Answer and Explanation is :

The correct answer is: ( \frac{4}{33} )

To express the repeating decimal ( 0.12\overline{12} ) as a fraction, we can follow a systematic approach using algebra.

1. Define the Decimal: Let ( x = 0.12\overline{12} ). This means ( x = 0.12121212… )
2. Eliminate the Repetition: To isolate the repeating part, we multiply both sides of the equation by 100 (since the repeat cycle has two digits). Thus:
   [
   100x = 12.121212…
   ]
3. Set Up the Equation: We now have two equations:
   [
   x = 0.121212…
   ]
   [
   100x = 12.121212…
   ]
4. Subtract the Two Equations: Subtract the first equation from the second to eliminate the repeating part:
   [
   100x – x = 12.121212… – 0.121212…
   ]
   This simplifies to:
   [
   99x = 12
   ]
5. Solve for ( x ): Divide both sides by 99:
   [
   x = \frac{12}{99}
   ]
6. Simplify the Fraction: We can simplify ( \frac{12}{99} ) by finding the greatest common divisor (GCD) of 12 and 99, which is 3. Divide both the numerator and denominator by 3:
   [
   x = \frac{12 \div 3}{99 \div 3} = \frac{4}{33}
   ]
7. Final Answer: Thus, ( 0.12\overline{12} ) can be expressed as the fraction ( \frac{4}{33} ).

In summary, the process involves defining the repeating decimal, multiplying to eliminate the repetition, forming a solvable equation, and then simplifying the result. The final simplified fraction representing ( 0.12\overline{12} ) is ( \frac{4}{33} ). This method can be applied to any repeating decimal to convert it into a fraction, making it a useful technique in mathematics.