Express each of the following recurring decimals into the rational number :
`(i)0.bar(32)" "(ii)0.bar(56)" "(iii)3.bar(18)" "(iv)10.bar(13)`

Explanation

To express the recurring decimals into rational numbers, we will follow a systematic approach for each case. Let's solve each part step by step.

### (i) Convert \(0.\overline{32}\) to a rational number:

1. **Let \( x = 0.\overline{32} \)**.
2. **Multiply both sides by 100** (since the repeating part has 2 digits):
\[ 100x = 32.\overline{32} \]
3. **Subtract the original equation from this new equation**:
\[ 100x - x = 32.\overline{32} - 0.\overline{32} \]
This simplifies to:
\[ 99x = 32 \]
4. **Solve for \( x \)**:
\[ x = \frac{32}{99} \]

### (ii) Convert \(0.\overline{56}\) to a rational number:

1. **Let \( x = 0.\overline{56} \)**.
2. **Multiply both sides by 100**:
\[ 100x = 56.\overline{56} \]
3. **Subtract the original equation from this new equation**:
\[ 100x - x = 56.\overline{56} - 0.\overline{56} \]
This simplifies to:
\[ 99x = 56 \]
4. **Solve for \( x \)**:
\[ x = \frac{56}{99} \]

### (iii) Convert \(3.\overline{18}\) to a rational number:

1. **Let \( x = 3.\overline{18} \)**.
2. **Multiply both sides by 100**:
\[ 100x = 318.\overline{18} \]
3. **Subtract the original equation from this new equation**:
\[ 100x - x = 318.\overline{18} - 3.\overline{18} \]
This simplifies to:
\[ 99x = 315 \]
4. **Solve for \( x \)**:
\[ x = \frac{315}{99} \]

### (iv) Convert \(10.\overline{13}\) to a rational number:

1. **Let \( x = 10.\overline{13} \)**.
2. **Multiply both sides by 100**:
\[ 100x = 1013.\overline{13} \]
3. **Subtract the original equation from this new equation**:
\[ 100x - x = 1013.\overline{13} - 10.\overline{13} \]
This simplifies to:
\[ 99x = 1003 \]
4. **Solve for \( x \)**:
\[ x = \frac{1003}{99} \]

### Summary of Results:
- (i) \(0.\overline{32} = \frac{32}{99}\)
- (ii) \(0.\overline{56} = \frac{56}{99}\)
- (iii) \(3.\overline{18} = \frac{315}{99}\)
- (iv) \(10.\overline{13} = \frac{1003}{99}\)