Question

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

0.523523... = what fraction?

(5 points)

x = 523/999
999x = 523
Let x = 0.523523
1000x = 523.523523
Subtract 1x = 0.523523

Answer 1

To convert the repeating decimal \(0.523523...\) to a fraction, the steps can be ordered as follows:

1. Let \( x = 0.523523... \) (This sets up the equation)
2. Multiply both sides by 1000 to shift the decimal point three places: \( 1000x = 523.523523... \)
3. Now subtract the original \( x \) from this equation: \( 1000x - x = 523.523523... - 0.523523... \) which simplifies to \( 999x = 523 \)
4. Now solve for \( x \): \( x = \frac{523}{999} \)

So the correct order is:
1. Let \( x = 0.523523 \)
2. \( 1000x = 523.523523 \)
3. Subtract \( 1x = 0.523523 \)
4. \( 999x = 523 \)
5. \( x = \frac{523}{999} \)

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