Question

Write the following infinite decimal expansion as a fraction. You do not need to simplify.


0.555...

Answer 1

To convert the infinite decimal expansion \(0.555...\) into a fraction, we can use the following method:

Let \(x\) be equal to the decimal:

\[
x = 0.555...
\]

Next, we multiply both sides of the equation by 10:

\[
10x = 5.555...
\]

Now, observe that we can rewrite \(5.555...\) in terms of \(x\):

\[
10x = 5 + 0.555... = 5 + x
\]

Now, we can set up the equation:

\[
10x = 5 + x
\]

To isolate \(x\), subtract \(x\) from both sides:

\[
10x - x = 5
\]

This simplifies to:

\[
9x = 5
\]

Now, divide both sides by 9:

\[
x = \frac{5}{9}
\]

Thus, the infinite decimal expansion \(0.555...\) can be expressed as the fraction:

\[
\frac{5}{9}
\]