Express \(0.6 + 0.\overline{7} + 0.4\overline{7}\) in the Form \( \frac{p}{q} \)
Question: Express \(0.6 + 0.\overline{7} + 0.4\overline{7}\) (bar on 7) in the form \( \frac{p}{q} \), where \(q \ne 0\).
Solution:
Step 1: Convert each decimal into fraction
\[ 0.6 = \frac{6}{10} = \frac{3}{5} \]
\[ 0.\overline{7} = \frac{7}{9} \]
\[ 0.4\overline{7} = \frac{43}{90} \]
Step 2: Add all fractions
\[ \frac{3}{5} + \frac{7}{9} + \frac{43}{90} \]
LCM of 5, 9, 90 = 90
\[ \frac{3}{5} = \frac{54}{90}, \quad \frac{7}{9} = \frac{70}{90}, \quad \frac{43}{90} = \frac{43}{90} \]
\[ \frac{54 + 70 + 43}{90} = \frac{167}{90} \]
Final Answer:
\[ 0.6 + 0.\overline{7} + 0.4\overline{7} = \frac{167}{90} \]
Concept Used:
Convert each decimal into fraction form and then add using LCM of denominators.