Express the Decimal \(0.\overline{621}\) in the Form \( \frac{p}{q} \)
Question: Express \(0.\overline{621}\) in the form \( \frac{p}{q} \).
Solution:
Let
\[ x = 0.\overline{621} \]
Multiply both sides by 1000 (since three digits repeat):
\[ 1000x = 621.\overline{621} \]
Subtract the first equation from the second:
\[ 1000x – x = 621.\overline{621} – 0.\overline{621} \]
\[ 999x = 621 \]
\[ x = \frac{621}{999} \]
Simplify by dividing numerator and denominator by 27:
\[ \frac{621}{999} = \frac{23}{37} \]
Final Answer:
\[ 0.\overline{621} = \frac{23}{37} \]
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.