Express the Decimal \(125.\overline{3}\) in the Form \( \frac{p}{q} \)
Question: Express \(125.\overline{3}\) in the form \( \frac{p}{q} \).
Solution:
Let
\[ x = 125.\overline{3} \]
Multiply both sides by 10 (since one digit repeats):
\[ 10x = 1253.\overline{3} \]
Subtract the first equation from the second:
\[ 10x – x = 1253.\overline{3} – 125.\overline{3} \]
\[ 9x = 1128 \]
\[ x = \frac{1128}{9} \]
Simplify:
\[ \frac{1128}{9} = \frac{376}{3} \]
Final Answer:
\[ 125.\overline{3} = \frac{376}{3} \]
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.