Express the Decimal \(0.\overline{4}\) in the Form \( \frac{p}{q} \)
Question: Express \(0.\overline{4}\) in the form \( \frac{p}{q} \).
Solution:
Let
\[ x = 0.\overline{4} \]
Multiply both sides by 10:
\[ 10x = 4.\overline{4} \]
Subtract the first equation from the second:
\[ 10x – x = 4.\overline{4} – 0.\overline{4} \]
\[ 9x = 4 \]
\[ x = \frac{4}{9} \]
Final Answer:
\[ 0.\overline{4} = \frac{4}{9} \]
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.