To express the repeating decimal [tex]\(2 . \overline{4}\)[/tex] as a fraction in simplest form, we start by setting the repeating decimal equal to a variable:
Let [tex]\( x = 2 . \overline{4} \)[/tex].
First, we want to eliminate the repeating part by manipulating the equation.
Multiply both sides of the equation by 10 (since the repetend is a single digit), which shifts the decimal point one place to the right:
[tex]\[
10x = 24.4\overline{4}
\][/tex]
Now we have two equations:
[tex]\[
x = 2.4\overline{4}
\][/tex]
[tex]\[
10x = 24.4\overline{4}
\][/tex]
Next, subtract the first equation from the second:
[tex]\[
10x - x = 24.4\overline{4} - 2.4\overline{4}
\][/tex]
This simplifies to:
[tex]\[
9x = 22
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = \frac{22}{9}
\][/tex]
Thus, the repeating decimal [tex]\( 2 . \overline{4} \)[/tex] simplifies to the mixed number:
[tex]\[
2 + \frac{22}{9}
\][/tex]
[tex]\[
= 2 \frac{22}{9}
\][/tex]
However, it's often given in a more compact form in the context of four given choices, and none of those are improper fractions.
So, we compare this result to the provided options:
- [tex]\(2 \frac{1}{4} = 2.25\)[/tex]
- [tex]\(2 \frac{2}{5} = 2.4\)[/tex]
- [tex]\(2 \frac{11}{25} = 2.44\)[/tex]
- [tex]\(2 \frac{4}{9} = 2.44444\)[/tex]
From the above options, it is clear that [tex]\(2 \frac{4}{9} = 2.44444\)[/tex], which matches our result.
Therefore, the correct expression for [tex]\(2 . \overline{4}\)[/tex] as a fraction in simplest form is:
[tex]\[
2 \frac{4}{9}
\][/tex]
