To express [tex]\(2.417 \overline{8}\)[/tex] in the form [tex]\(\frac{a}{b}\)[/tex], we start by closely examining the given number. The number [tex]\(2.417 \overline{8}\)[/tex] can be written as a sum of a non-repeating part and a repeating decimal part.
First, let's set [tex]\(x = 2.417 \overline{8}\)[/tex], which means [tex]\(x = 2.4178888\ldots\)[/tex].
### Step 1: Decompose the Number
Let's decompose [tex]\(x\)[/tex] into two parts: [tex]\(2.417\)[/tex] and [tex]\(0.0008888\ldots\)[/tex].
1. Non-repeating Part:
The non-repeating part is [tex]\(2.417\)[/tex]. This can be written as:
[tex]\[
2.417 = \frac{2417}{1000}
\][/tex]
2. Repeating Part:
Now let's isolate the repeating part of the decimal, which is [tex]\(0.0008\overline{8}\)[/tex].
### Step 2: Express the Repeating Part as a Fraction
Let [tex]\(y = 0.0008\overline{8}\)[/tex]. To simplify this, let's write it in a form that isolates the repeating decimal.
[tex]\[
z = 0.\overline{8}
\][/tex]
By definition, the repeating decimal [tex]\(0.\overline{8}\)[/tex] is:
[tex]\[
z = 0.8888\ldots
\][/tex]
We can express [tex]\(z\)[/tex] as a fraction by solving the equation:
[tex]\[
z = 0.\overline{8}
\][/tex]
Multiply both sides by 10:
[tex]\[
10z = 8.\overline{8}
\][/tex]
Subtract the original equation from this new one:
[tex]\[
10z - z = 8.\overline{8} - 0.\overline{8}
\][/tex]
This simplifies to:
[tex]\[
9z = 8
\][/tex]
So,
[tex]\[
z = \frac{8}{9}
\][/tex]
Since [tex]\(y = 0.0008\overline{8}\)[/tex], we can rewrite [tex]\(y\)[/tex] as:
[tex]\[
y = 0.0008z = 0.0008 \times \frac{8}{9} = \frac{8}{10000} \times \frac{8}{9} = \frac{8 \times 8}{10000 \times 9} = \frac{64}{90000} = \frac{8}{9990}
\][/tex]
### Step 3: Combining Both Parts
Now we combine the two parts, [tex]\(2.417\)[/tex] and [tex]\(0.0008 \overline{8}\)[/tex]:
[tex]\[
x = 2.417 + 0.0008 \overline{8} = \frac{2417}{1000} + \frac{8}{9990}
\][/tex]
### Step 4: Find a Common Denominator and Add
To add these fractions, we need a common denominator. The least common multiple (LCM) of 1000 and 9990 is 4995000.
Convert each fraction to have the common denominator:
[tex]\[
\frac{2417}{1000} \times \frac{4995}{4995} = \frac{12067815}{4995000}
\][/tex]
[tex]\[
\frac{8}{9990} \times \frac{499500}{499500} = \frac{4000}{4995000}
\][/tex]
Adding these fractions gives:
[tex]\[
\frac{12067815 + 4000}{4995000} = \frac{12071815}{4995000}
\][/tex]
Thus,
[tex]\[
x = \frac{12071815}{4995000}
\][/tex]
Therefore, [tex]\(2.417 \overline{8}\)[/tex] expressed in the form [tex]\(\frac{a}{b}\)[/tex] is:
[tex]\[
\boxed{\frac{12071815}{4995000}}
\][/tex]
