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 convert the given recurring decimals into rational numbers, we will follow a systematic approach for each case.

### (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 first equation from the second**:
\[ 100x - x = 32.\overline{32} - 0.\overline{32} \]
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 first equation from the second**:
\[ 100x - x = 56.\overline{56} - 0.\overline{56} \]
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 first equation from the second**:
\[ 100x - x = 318.\overline{18} - 3.\overline{18} \]
This simplifies to:
\[ 99x = 315 \]
4. **Solve for \(x\)**:
\[ x = \frac{315}{99} \]
We can simplify this fraction:
\[ x = \frac{105}{33} = \frac{35}{11} \]

### (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 first equation from the second**:
\[ 100x - x = 1013.\overline{13} - 10.\overline{13} \]
This simplifies to:
\[ 99x = 1003 \]
4. **Solve for \(x\)**:
\[ x = \frac{1003}{99} \]

### Summary of Results
1. \(0.\overline{32} = \frac{32}{99}\)
2. \(0.\overline{56} = \frac{56}{99}\)
3. \(3.\overline{18} = \frac{35}{11}\)
4. \(10.\overline{13} = \frac{1003}{99}\)