Express the Decimal \(0.\overline{37}\) in the Form \( \frac{p}{q} \)
Question: Express \(0.\overline{37}\) in the form \( \frac{p}{q} \).
Solution:
Let
\[ x = 0.\overline{37} \]
Multiply both sides by 100 (since two digits are repeating):
\[ 100x = 37.\overline{37} \]
Subtract the first equation from the second:
\[ 100x – x = 37.\overline{37} – 0.\overline{37} \]
\[ 99x = 37 \]
\[ x = \frac{37}{99} \]
Final Answer:
\[ 0.\overline{37} = \frac{37}{99} \]
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.