Express each of the following as a fraction in simplest form :
` 0.overline(8)` (ii)` 2.overline(4)` (iii) ` 0.overline(24)` (iv) ` 0.1overline2`
(vi) 0.00608

Answer

To express each of the given numbers as a fraction in simplest form, we will follow a systematic approach for each number. Let's go through them one by one.

### (i) Express \( 0.\overline{8} \) as a fraction:

1. **Let \( x = 0.\overline{8} \)**.
2. **Multiply both sides by 10**:
\[ 10x = 8.\overline{8} \]
3. **Subtract the first equation from the second**:
\[ 10x - x = 8.\overline{8} - 0.\overline{8} \]
This simplifies to:
\[ 9x = 8 \]
4. **Solve for \( x \)**:
\[ x = \frac{8}{9} \]

### (ii) Express \( 2.\overline{4} \) as a fraction:

1. **Let \( x = 2.\overline{4} \)**.
2. **Multiply both sides by 10**:
\[ 10x = 24.\overline{4} \]
3. **Subtract the first equation from the second**:
\[ 10x - x = 24.\overline{4} - 2.\overline{4} \]
This simplifies to:
\[ 9x = 22 \]
4. **Solve for \( x \)**:
\[ x = \frac{22}{9} \]

### (iii) Express \( 0.\overline{24} \) as a fraction:

1. **Let \( x = 0.\overline{24} \)**.
2. **Multiply both sides by 100** (since 24 has two digits):
\[ 100x = 24.\overline{24} \]
3. **Subtract the first equation from the second**:
\[ 100x - x = 24.\overline{24} - 0.\overline{24} \]
This simplifies to:
\[ 99x = 24 \]
4. **Solve for \( x \)**:
\[ x = \frac{24}{99} \]
5. **Simplify \( \frac{24}{99} \)**:
\[ x = \frac{8}{33} \]

### (iv) Express \( 0.1\overline{2} \) as a fraction:

1. **Let \( x = 0.1\overline{2} \)**.
2. **Multiply both sides by 10**:
\[ 10x = 1.\overline{2} \]
3. **Multiply both sides by 10 again**:
\[ 100x = 12.\overline{2} \]
4. **Subtract the first equation from the second**:
\[ 100x - 10x = 12.\overline{2} - 1.\overline{2} \]
This simplifies to:
\[ 90x = 11 \]
5. **Solve for \( x \)**:
\[ x = \frac{11}{90} \]

### (v) Express \( 0.00608 \) as a fraction:

1. **Write it as a fraction**:
\[ 0.00608 = \frac{608}{100000} \]
2. **Simplify \( \frac{608}{100000} \)**:
- Find the GCD of 608 and 100000. The GCD is 16.
- Divide both the numerator and denominator by 16:
\[ \frac{608 \div 16}{100000 \div 16} = \frac{38}{6250} \]

### Summary of Results:
1. \( 0.\overline{8} = \frac{8}{9} \)
2. \( 2.\overline{4} = \frac{22}{9} \)
3. \( 0.\overline{24} = \frac{8}{33} \)
4. \( 0.1\overline{2} = \frac{11}{90} \)
5. \( 0.00608 = \frac{38}{6250} \)