Express each of the following recurring decimals into the rational number :
`(i)0.bar(7)" "(ii)0.bar(6)" "(iii)1.bar(3)" "(iv)3.bar(8)`

Explanation

Let's solve each of the recurring decimals step by step:

### (i) Convert \(0.\overline{7}\) to a rational number

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

2. **Multiply both sides by 10:**
\(10x = 7.77777...\)

3. **Subtract the first equation from the second:**
\(10x - x = 7.77777... - 0.77777...\)
This simplifies to:
\(9x = 7\)

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

### (ii) Convert \(0.\overline{6}\) to a rational number

1. **Let \(x = 0.\overline{6}\)**
This means \(x = 0.66666...\)

2. **Multiply both sides by 10:**
\(10x = 6.66666...\)

3. **Subtract the first equation from the second:**
\(10x - x = 6.66666... - 0.66666...\)
This simplifies to:
\(9x = 6\)

4. **Solve for \(x\):**
\(x = \frac{6}{9} = \frac{2}{3}\)

### (iii) Convert \(1.\overline{3}\) to a rational number

1. **Let \(x = 1.\overline{3}\)**
This means \(x = 1.33333...\)

2. **Multiply both sides by 10:**
\(10x = 13.33333...\)

3. **Subtract the first equation from the second:**
\(10x - x = 13.33333... - 1.33333...\)
This simplifies to:
\(9x = 12\)

4. **Solve for \(x\):**
\(x = \frac{12}{9} = \frac{4}{3}\)

### (iv) Convert \(3.\overline{8}\) to a rational number

1. **Let \(x = 3.\overline{8}\)**
This means \(x = 3.88888...\)

2. **Multiply both sides by 10:**
\(10x = 38.88888...\)

3. **Subtract the first equation from the second:**
\(10x - x = 38.88888... - 3.88888...\)
This simplifies to:
\(9x = 35\)

4. **Solve for \(x\):**
\(x = \frac{35}{9}\)

### Summary of Results:
- \(0.\overline{7} = \frac{7}{9}\)
- \(0.\overline{6} = \frac{2}{3}\)
- \(1.\overline{3} = \frac{4}{3}\)
- \(3.\overline{8} = \frac{35}{9}\)