Express the Decimal \(4.\overline{7}\) in the Form \( \frac{p}{q} \)
Question: Express \(4.\overline{7}\) in the form \( \frac{p}{q} \).
Solution:
Let
\[ x = 4.\overline{7} \]
Multiply both sides by 10 (since one digit repeats):
\[ 10x = 47.\overline{7} \]
Subtract the first equation from the second:
\[ 10x – x = 47.\overline{7} – 4.\overline{7} \]
\[ 9x = 43 \]
\[ x = \frac{43}{9} \]
Final Answer:
\[ 4.\overline{7} = \frac{43}{9} \]
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.