Find a rational number for the following which will have as its expantion :
(i) `0.6bar8` (ii) `0.23bar4`

Answer

To find the rational numbers corresponding to the decimal expansions \(0.6\overline{8}\) and \(0.23\overline{4}\), we will follow a systematic approach for each part.

### Part (i): Finding the rational number for \(0.6\overline{8}\)

1. **Express the repeating decimal**:
Let \(x = 0.68888...\) (where \(8\) repeats indefinitely).

2. **Multiply by a power of 10 to shift the decimal point**:
Since the repeating part has one digit, multiply by \(10\):
\[ 10x = 6.8888... \]

3. **Set up the equation**:
Now, we have:
\[ 10x = 6.8888... \]
\[ x = 0.6888... \]

4. **Subtract the two equations**:
Subtract the second equation from the first:
\[ 10x - x = 6.8888... - 0.6888... \]
This simplifies to:
\[ 9x = 6.2 \]

5. **Solve for \(x\)**:
\[ x = \frac{6.2}{9} \]
To convert \(6.2\) to a fraction:
\[ 6.2 = \frac{62}{10} = \frac{31}{5} \]
Thus,
\[ x = \frac{31/5}{9} = \frac{31}{45} \]

### Part (ii): Finding the rational number for \(0.23\overline{4}\)

1. **Express the repeating decimal**:
Let \(y = 0.23444...\) (where \(4\) repeats indefinitely).

2. **Multiply by a power of 10**:
Since the repeating part has one digit, multiply by \(10\):
\[ 10y = 2.3444... \]

3. **Set up the equation**:
Now, we have:
\[ 10y = 2.3444... \]
\[ y = 0.2344... \]

4. **Subtract the two equations**:
Subtract the second equation from the first:
\[ 10y - y = 2.3444... - 0.2344... \]
This simplifies to:
\[ 9y = 2.11 \]

5. **Solve for \(y\)**:
\[ y = \frac{2.11}{9} \]
To convert \(2.11\) to a fraction:
\[ 2.11 = \frac{211}{100} \]
Thus,
\[ y = \frac{211/100}{9} = \frac{211}{900} \]

### Final Answers:
- For \(0.6\overline{8}\), the rational number is \(\frac{31}{45}\).
- For \(0.23\overline{4}\), the rational number is \(\frac{211}{900}\).