Question

Put the steps in order for changing the repeating decimal, which is rational, to a ratio or fraction
0.523523...= what fraction?

let x=0.523523...
subtract 1x=0.523523...
x=523/999
999x=523
1000x=523.523523...

Answer 1

To convert the repeating decimal \(0.523523...\) into a fraction, let's put the steps in proper order:

1. **Let \( x = 0.523523...\)** - This is our starting point where we define \( x \) as the repeating decimal.

2. **Multiply by a power of 10** - Since the repeating part has 3 digits ("523"), we multiply both sides of the equation by \( 1000 \) (which shifts the decimal point three places to the right):
\[
1000x = 523.523523...
\]

3. **Set up the equation** - Now we have the equation:
\[
1000x = 523 + x
\]

4. **Subtract \( x \) from both sides** - This helps to eliminate \( x \) from the right side:
\[
1000x - x = 523
\]
which simplifies to:
\[
999x = 523
\]

5. **Solve for \( x \)** - Now, divide both sides by 999 to find \( x \):
\[
x = \frac{523}{999}
\]

Thus, the final fraction representation of the repeating decimal \( 0.523523... \) is \(\frac{523}{999}\).