Certainly! Let's convert the repeating decimal [tex]\( 0.\overline{51} \)[/tex] into a fraction.
1. Let [tex]\( x \)[/tex] be the repeating decimal number:
[tex]\[
x = 0.515151\ldots
\][/tex]
2. To eliminate the repeating decimal, multiply [tex]\( x \)[/tex] by a power of 10 (in this case, 100) to shift the decimal point to the right by two places:
[tex]\[
100x = 51.515151\ldots
\][/tex]
3. Now, set up an equation with the repeating decimal:
[tex]\[
x = 0.515151\ldots \quad \text{(Equation 1)}
\][/tex]
[tex]\[
100x = 51.515151\ldots \quad \text{(Equation 2)}
\][/tex]
4. Subtract Equation 1 from Equation 2 to eliminate the repeating part:
[tex]\[
100x - x = 51.515151\ldots - 0.515151\ldots
\][/tex]
[tex]\[
99x = 51
\][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 99:
[tex]\[
x = \frac{51}{99}
\][/tex]
6. Simplify the fraction [tex]\( \frac{51}{99} \)[/tex]. To do this, find the greatest common divisor (GCD) of the numerator (51) and the denominator (99).
- The GCD of 51 and 99 is 3.
Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{51 \div 3}{99 \div 3} = \frac{17}{33}
\][/tex]
So, the repeating decimal [tex]\( 0.\overline{51} \)[/tex] can be written as the fraction [tex]\( \frac{17}{33} \)[/tex].
7. Finally, converting the simplified fraction back to a decimal to check:
[tex]\[
\frac{17}{33} = 0.5151515151515151\ldots
\][/tex]
Hence, the repeating decimal [tex]\( 0.\overline{51} \)[/tex] is equivalent to the fraction [tex]\( \frac{17}{33} \)[/tex], and the decimal value accurately represents this fraction.
