Question

Convert 0.8 the bars above the eight to a rational number

Answer 1

To convert the repeating decimal \(0.8\overline{8}\) (where the bar indicates that the 8 repeats indefinitely) into a rational number, we can follow these steps:

Let \(x = 0.8\overline{8}\).

This means that:

\[ x = 0.88888\ldots \]

Now, to eliminate the repeating part, we can multiply \(x\) by 10:

\[ 10x = 8.88888\ldots \]

Now we have two equations:

1. \(x = 0.88888\ldots\)
2. \(10x = 8.88888\ldots\)

Next, we can subtract the first equation from the second:

\[
10x - x = 8.88888\ldots - 0.88888\ldots
\]

This simplifies to:

\[
9x = 8
\]

Now, solving for \(x\):

\[
x = \frac{8}{9}
\]

Therefore, the rational number representation of \(0.8\overline{8}\) is

\[
\frac{8}{9}.
\]