Express the Decimal \(0.4\overline{7}\) in the Form \( \frac{p}{q} \)
Question: Express \(0.4\overline{7}\) (bar only on 7) in the form \( \frac{p}{q} \).
Solution:
Let
\[ x = 0.4\overline{7} \]
Multiply by 10 (to move non-repeating part):
\[ 10x = 4.\overline{7} \]
Now multiply by 10 again (since one digit repeats):
\[ 100x = 47.\overline{7} \]
Subtract the two equations:
\[ 100x – 10x = 47.\overline{7} – 4.\overline{7} \]
\[ 90x = 43 \]
\[ x = \frac{43}{90} \]
Final Answer:
\[ 0.4\overline{7} = \frac{43}{90} \]
Concept Used:
For mixed recurring decimals, first eliminate the non-repeating part, then eliminate the repeating part using subtraction.