Express the Decimal \(0.\overline{54}\) in the Form \( \frac{p}{q} \)
Question: Express \(0.\overline{54}\) in the form \( \frac{p}{q} \).
Solution:
Let
\[ x = 0.\overline{54} \]
Multiply both sides by 100 (since two digits repeat):
\[ 100x = 54.\overline{54} \]
Subtract the first equation from the second:
\[ 100x – x = 54.\overline{54} – 0.\overline{54} \]
\[ 99x = 54 \]
\[ x = \frac{54}{99} \]
Simplify by dividing numerator and denominator by 9:
\[ \frac{54}{99} = \frac{6}{11} \]
Final Answer:
\[ 0.\overline{54} = \frac{6}{11} \]
Concept Used:
To convert a recurring decimal into a fraction, assume it as a variable, multiply by a suitable power of 10, and subtract to eliminate repeating digits.