To convert the recurring decimal [tex]\(4.\overline{562}\)[/tex] into a fraction, we need to express it in terms of non-repeating and repeating parts. Let's break down the steps:
### Step 1: Identify the components
The given repeating decimal is [tex]\(4.\overline{562}\)[/tex]. This can be broken down into:
- The integer part: 4.
- The repeating decimal part: [tex]\(\overline{562}\)[/tex].
### Step 2: Represent the repeating part as a fraction
First, consider the repeating part [tex]\(\overline{562}\)[/tex].
Let [tex]\( x = 0.\overline{562} \)[/tex]. This means:
[tex]\[ x = 0.562562562\ldots \][/tex]
To eliminate the repeating part, multiply [tex]\( x \)[/tex] by 1000 (since the repeating part is 3 digits long):
[tex]\[ 1000x = 562.562562\ldots \][/tex]
Now, subtract the initial [tex]\( x \)[/tex] from this equation:
[tex]\[ 1000x - x = 562.562562\ldots - 0.562562\ldots \][/tex]
[tex]\[ 999x = 562 \][/tex]
[tex]\[ x = \frac{562}{999} \][/tex]
### Step 3: Combine the integer part and the fraction
The entire decimal [tex]\(4.\overline{562}\)[/tex] can now be written as the sum of the integer part and the fraction part:
[tex]\[ 4 + 0.\overline{562} = 4 + \frac{562}{999} \][/tex]
To combine these into a single fraction:
[tex]\[ 4 + \frac{562}{999} = \frac{4 \cdot 999 + 562}{999} = \frac{3996 + 562}{999} = \frac{4558}{999} \][/tex]
### Step 4: Identify the correct option
The fraction form of the recurring decimal [tex]\(4.\overline{562}\)[/tex] is [tex]\(\frac{4558}{999}\)[/tex].
Therefore, the best answer is:
[tex]\[ \boxed{C. \frac{4558}{999}} \][/tex]
