Express the Decimal \(0.12\overline{3}\) in the Form \( \frac{p}{q} \)
Question: Express \(0.12\overline{3}\) (bar only on 3) in the form \( \frac{p}{q} \).
Solution:
Let
\[ x = 0.12\overline{3} \]
Multiply by 100 (to remove non-repeating part):
\[ 100x = 12.\overline{3} \]
Now multiply by 10 (since one digit repeats):
\[ 1000x = 123.\overline{3} \]
Subtract the two equations:
\[ 1000x – 100x = 123.\overline{3} – 12.\overline{3} \]
\[ 900x = 111 \]
\[ x = \frac{111}{900} \]
Simplify:
\[ \frac{111}{900} = \frac{37}{300} \]
Final Answer:
\[ 0.12\overline{3} = \frac{37}{300} \]
Concept Used:
For mixed recurring decimals, first eliminate the non-repeating digits, then eliminate the repeating part by subtraction.