Evaluate `.23bar(45)`.

Answer

To evaluate the repeating decimal \(0.23\overline{45}\), we can follow these steps:

### Step 1: Express the repeating decimal
We can express \(0.23\overline{45}\) as:
\[ 0.23\overline{45} = 0.235454545\ldots \]
This means that the digits "45" repeat indefinitely.

### Step 2: Split the decimal into two parts
We can separate the non-repeating and repeating parts:
\[ 0.23 + 0.00\overline{45} \]

### Step 3: Convert the non-repeating part
The non-repeating part \(0.23\) can be expressed as a fraction:
\[ 0.23 = \frac{23}{100} \]

### Step 4: Convert the repeating part
Now, we need to convert the repeating decimal \(0.00\overline{45}\) into a fraction. Let:
\[ x = 0.00\overline{45} \]
Then, multiplying both sides by 100 (to shift the decimal point two places to the right):
\[ 100x = 4.545454\ldots \]
Now, we can also express \(4.545454\ldots\) as:
\[ 4.545454\ldots = 4 + 0.545454\ldots \]
Let \(y = 0.545454\ldots\). Then:
\[ 100y = 54.545454\ldots \]
Subtracting the original \(y\) from this equation:
\[ 100y - y = 54.545454\ldots - 0.545454\ldots \]
This simplifies to:
\[ 99y = 54 \implies y = \frac{54}{99} = \frac{6}{11} \]
Thus, we have:
\[ x = 4 + \frac{6}{11} = \frac{44}{11} + \frac{6}{11} = \frac{50}{11} \]

### Step 5: Combine the two parts
Now we can combine the two parts:
\[ 0.23\overline{45} = \frac{23}{100} + \frac{50}{11} \]

### Step 6: Find a common denominator
The least common multiple of 100 and 11 is 1100. We convert both fractions:
\[ \frac{23}{100} = \frac{23 \times 11}{100 \times 11} = \frac{253}{1100} \]
\[ \frac{50}{11} = \frac{50 \times 100}{11 \times 100} = \frac{5000}{1100} \]

### Step 7: Add the fractions
Now we can add the two fractions:
\[ 0.23\overline{45} = \frac{253}{1100} + \frac{5000}{1100} = \frac{5253}{1100} \]

### Final Result
Thus, the value of \(0.23\overline{45}\) is:
\[ \frac{5253}{1100} \]