Question

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

0.523523... = what fraction?

1000x = 532.523523...
Subtract 1x = 0.523523
Let x = 0.523523
999x = 523
x = 523/999

Answer 1

To convert the repeating decimal \( 0.523523... \) into a fraction, you can follow these ordered steps:

1. **Let \( x = 0.523523... \)** (Define the repeating decimal as a variable \( x \)).

2. **Multiply by 1000** to shift the decimal point three places right (since the repeating part has 3 digits):
\[
1000x = 523.523523...
\]

3. **Subtract the original \( x \) from the equation**:
\[
1000x - 1x = 523.523523... - 0.523523...
\]
Which simplifies to:
\[
999x = 523
\]

4. **Solve for \( x \)** by dividing both sides by 999:
\[
x = \frac{523}{999}
\]

So, the steps in order are:

1. Let \( x = 0.523523... \)
2. Multiply by 1000: \( 1000x = 523.523523... \)
3. Subtract \( 1x \): \( 999x = 523 \)
4. Solve for \( x \): \( x = \frac{523}{999} \)

Thus, \( 0.523523... = \frac{523}{999} \).