Question

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



0.418418418...0.418418418...0.418418418

Answer 1

−2x=44

Answer 2

To express the repeating decimal \( 0.418418418\ldots \) as a fraction, we can denote it as \( x \):

\[
x = 0.418418418\ldots
\]

The repeating portion "418" has a length of 3 digits. To eliminate the repeating decimal, we will multiply \( x \) by \( 1000 \) (since the repeating section is 3 digits long):

\[
1000x = 418.418418418\ldots
\]

Now we can set up an equation using the two expressions we have for \( x \):

\[
1000x = 418 + x
\]

Next, we rearrange the equation to isolate \( x \):

\[
1000x - x = 418
\]
\[
999x = 418
\]

Now, we can solve for \( x \):

\[
x = \frac{418}{999}
\]

Thus, the infinite decimal expansion \( 0.418418418\ldots \) can be expressed as the fraction:

\[
\frac{418}{999}
\]

You do not need to simplify this fraction further, so the final answer is:

\[
\frac{418}{999}
\]

Answer 3

To solve the equation \(-2x = 44\), follow these steps:

1. Divide both sides by \(-2\) to isolate \(x\):

\[
x = \frac{44}{-2}
\]

2. Simplify the right side:

\[
x = -22
\]

So the solution to the equation \(-2x = 44\) is:

\[
x = -22
\]