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 \( 0.555... \) into a fraction, we can let \( x \) represent the repeating decimal:

\[
x = 0.555...
\]

Next, we can multiply both sides of the equation by 10 to shift the decimal point one place to the right:

\[
10x = 5.555...
\]

Now we have two equations:

1. \( x = 0.555... \)
2. \( 10x = 5.555... \)

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

\[
10x - x = 5.555... - 0.555...
\]

This simplifies to:

\[
9x = 5
\]

Now, we can solve for \( x \) by dividing both sides by 9:

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

Thus, the fraction representation of the infinite decimal \( 0.555... \) is

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