Express the following recurring decimal expansions in the form p/q , where p and q are integers and `q ne 0` .
(a) `0.bar(7)`
(b) `0.2bar(57)`

Answer

To express the recurring decimal expansions in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \), we will solve the two parts of the question step by step.

### (a) Convert \( 0.\overline{7} \) to a fraction

1. **Let \( x = 0.\overline{7} \)**.
- This means \( x = 0.77777... \)

2. **Multiply both sides by 10**:
\[ 10x = 7.77777... \]

3. **Now, subtract the first equation from the second**:
\[ 10x - x = 7.77777... - 0.77777... \]
This simplifies to:
\[ 9x = 7 \]

4. **Solve for \( x \)**:
\[ x = \frac{7}{9} \]

Thus, \( 0.\overline{7} = \frac{7}{9} \).

### (b) Convert \( 0.2\overline{57} \) to a fraction

1. **Let \( x = 0.2\overline{57} \)**.
- This means \( x = 0.25757575... \)

2. **Multiply both sides by 100**:
\[ 100x = 25.757575... \]

3. **Now, multiply both sides by 10**:
\[ 10x = 2.575757... \]

4. **Now we have two equations**:
- \( 100x = 25.757575... \) (Equation 1)
- \( 10x = 2.575757... \) (Equation 2)

5. **Subtract Equation 2 from Equation 1**:
\[ 100x - 10x = 25.757575... - 2.575757... \]
This simplifies to:
\[ 90x = 23.18 \]

6. **Solve for \( x \)**:
\[ x = \frac{23.18}{90} \]

7. **Convert \( 23.18 \) into a fraction**:
\[ 23.18 = \frac{2318}{100} \]
Therefore,
\[ x = \frac{2318}{9000} \]

8. **Simplify the fraction**:
- Divide both the numerator and denominator by their GCD (which is 2):
\[ x = \frac{1159}{4500} \]

Thus, \( 0.2\overline{57} = \frac{1159}{4500} \).

### Final Answers:
- (a) \( 0.\overline{7} = \frac{7}{9} \)
- (b) \( 0.2\overline{57} = \frac{1159}{4500} \)