To express the repeating decimal \(2 . \overline{4}\) as a fraction in its simplest form, let's follow these steps:
1. Let \( x \) be the repeating decimal.
[tex]\[ x = 2.4444\ldots \][/tex]
2. Rewrite the decimal with the repeating part isolated.
Here, the repeating part is \(0.\overline{4}\).
Let's express \( x \) such that:
[tex]\[ x = 2 + 0.\overline{4} \][/tex]
3. Determine the repeating decimal part as a fraction.
Let's handle \( 0.\overline{4} \) separately.
Let \( y = 0.\overline{4} \).
4. Set up an equation for \( y \).
[tex]\[ y = 0.4444\ldots \][/tex]
5. Multiply \( y \) by 10 to move the decimal point one place to the right.
[tex]\[ 10y = 4.4444\ldots \][/tex]
6. Subtract the original \( y \) from this new equation to eliminate the repeating part.
[tex]\[ 10y - y = 4.4444\ldots - 0.4444\ldots \][/tex]
[tex]\[ 9y = 4 \][/tex]
7. Solve for \( y \).
[tex]\[ y = \frac{4}{9} \][/tex]
Thus, \(0.\overline{4} = \frac{4}{9}\).
8. Combine the integer part and the fractional part together.
[tex]\[ x = 2 + \frac{4}{9} \][/tex]
9. Express \( x \) as a mixed number:
[tex]\[ x = 2 \frac{4}{9} \][/tex]
Now, we have \(2 . \overline{4}\) expressed as a fraction in simplest form:
[tex]\[ 2 \frac{4}{9} \][/tex]
So, among the given options, the correct fraction form of \(2 . \overline{4}\) is:
[tex]\[ \boxed{2 \frac{4}{9}} \][/tex]
