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

Question

0.523523... = what fraction?

options:

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, follow these steps in order:

1. Let \( x = 0.523523... \)

2. Multiply both sides by 1000 (because the repeating part has 3 digits):
   \( 1000x = 523.523523... \)

3. Subtract the original \( x \) from this equation:
   \( 1000x - 1x = 523.523523... - 0.523523... \)
   This simplifies to:
   \( 999x = 523 \)

4. Solve for \( x \):
   \( x = \frac{523}{999} \)

So the correct order is:

1. Let \( x = 0.523523 \)
2. Multiply by 1000: \( 1000x = 532.523523... \)
3. Subtract \( 1x \): \( 999x = 523 \)
4. Solve for \( x \): \( x = \frac{523}{999} \)

Final Fraction

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