Convert the following repeating decimal to a fraction in simplest form.

Question

., start overline, 4, end overline
.
4

Answer 1

To convert the repeating decimal \( 0.4444\ldots \) (which can be written as \( 0.\overline{4} \)) into a fraction, we can use the following method:

Let \( x = 0.\overline{4} \).

This means: \[ x = 0.4444\ldots \]

Next, we can multiply both sides of this equation by 10 to shift the decimal point to the right: \[ 10x = 4.4444\ldots \]

Now, we can set up a system of equations:

1. \( x = 0.4444\ldots \)
2. \( 10x = 4.4444\ldots \)

Now, if we subtract the first equation from the second equation: \[ 10x - x = 4.4444\ldots - 0.4444\ldots \] This simplifies to: \[ 9x = 4 \]

Now, we can solve for \( x \): \[ x = \frac{4}{9} \]

Thus, the repeating decimal \( 0.\overline{4} \) as a fraction in simplest form is: \[ \boxed{\frac{4}{9}} \]