To convert the repeating decimal [tex]\(0.\overline{51}\)[/tex] into a fraction, we'll follow a systematic approach:
1. Let [tex]\(x = 0.\overline{51}\)[/tex]:
This means [tex]\(x\)[/tex] is a repeating decimal where the digits '51' repeat indefinitely: [tex]\(x = 0.51515151...\)[/tex].
2. Multiply by 100 to shift the decimal point:
[tex]\[100x = 51.51515151...\][/tex]
This shifts the repeating portion to the right of the decimal point.
3. Subtract the original [tex]\(x\)[/tex] from this equation:
[tex]\[100x - x = 51.51515151... - 0.51515151...\][/tex]
Simplifying the left-hand side and the right-hand side, we get:
[tex]\[99x = 51\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[x = \frac{51}{99}\][/tex]
5. Simplify the fraction [tex]\(\frac{51}{99}\)[/tex]:
To simplify [tex]\(\frac{51}{99}\)[/tex], we find the greatest common divisor (GCD) of 51 and 99. The GCD of 51 and 99 is 3. Thus, we can divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{51 \div 3}{99 \div 3} = \frac{17}{33}
\][/tex]
6. Conclusion:
Hence, the simplified fraction for the repeating decimal [tex]\(0.\overline{51}\)[/tex] is:
[tex]\[
0.\overline{51} = \frac{17}{33}
\][/tex]
Therefore, the repeating decimal [tex]\(0.\overline{51}\)[/tex] converts to the fraction [tex]\(\frac{17}{33}\)[/tex].
