Question

Write the following infinite decimal expansion as a fraction. You do not need to simplify.
0.154154154

Answer 1

I don't exactly understand this, what's the answer?

Answer 2

To express the repeating decimal \(0.154154154...\) as a fraction, we can use the notation for repeating decimals.

Let's denote the repeating decimal as:
\[
x = 0.154154154...
\]

Now, observe that the repeating part "154" has three digits. To eliminate the repeating decimal, we can multiply \(x\) by \(1000\) (since "154" consists of three digits):
\[
1000x = 154.154154154...
\]

Next, we can set up an equation by subtracting the original \(x\) from this equation:
\[
1000x - x = 154.154154154... - 0.154154154...
\]
This simplifies to:
\[
999x = 154
\]

Now, we can solve for \(x\):
\[
x = \frac{154}{999}
\]

Thus, the infinite decimal expansion \(0.154154154...\) as a fraction is:
\[
\frac{154}{999}
\]

Answer 3

The repeating decimal \(0.154154154...\) can be written as the fraction \(\frac{154}{999}\).

There it is in a simple form:
\[
0.154154154... = \frac{154}{999}
\]

If you need further clarification on any steps, feel free to ask!