Express `2.3bar(45)` in the form of `p//q` where p and q are coprimes.

Answer

To express the repeating decimal \(2.3\overline{45}\) in the form of \(\frac{p}{q}\) where \(p\) and \(q\) are coprime, we can follow these steps:

### Step-by-Step Solution

1. **Let \(x\) be the repeating decimal**:
\[ x = 2.345454545\ldots \]

2. **Multiply by 10 to shift the decimal point**:
\[ 10x = 23.454545\ldots \]

3. **Multiply by 100 to shift the decimal point further**:
\[ 100x = 234.545454\ldots \]

4. **Set up the equations**:
From the above, we have:
- Equation 1: \(x = 2.345454545\ldots\)
- Equation 2: \(10x = 23.454545\ldots\)
- Equation 3: \(100x = 234.545454\ldots\)

5. **Subtract Equation 2 from Equation 3**:
\[ 100x - 10x = 234.545454\ldots - 23.454545\ldots \]
This simplifies to:
\[ 90x = 211.09 \]

6. **Solve for \(x\)**:
\[ x = \frac{211.09}{90} \]

7. **Convert the decimal in the numerator to a fraction**:
The decimal \(211.09\) can be expressed as:
\[ 211.09 = \frac{21109}{100} \]
Therefore,
\[ x = \frac{21109}{100} \cdot \frac{1}{90} = \frac{21109}{9000} \]

8. **Simplify the fraction**:
We need to check if \(21109\) and \(9000\) have any common factors. After checking, we find that they are coprime (no common factors other than 1).

9. **Final expression**:
Thus, we can express \(2.3\overline{45}\) as:
\[ x = \frac{21109}{9000} \]

### Final Answer:
\[ x = \frac{21109}{9000} \]