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

Question

0.523523... = what fraction?

(5 points)
responses
let x = 0.523523
x = 523/999
999x = 523
1000x = 523.523523
subtract 1x = 0.523523

Answer 1

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

1. Let \( x = 0.523523...\).
2. Multiply both sides by \( 1000 \) to shift the decimal places: \[ 1000x = 523.523523... \]
3. Recognizing that \( 0.523523...\) is equal to \( x \): \[ 1000x = 523 + x \]
4. Subtract \( x \) from both sides: \[ 999x = 523 \]
5. Solve for \( x \) by dividing both sides by \( 999 \): \[ x = \frac{523}{999} \]

So, the ordered steps are:

1. Let \( x = 0.523523...\).
2. Multiply \( x \) by \( 1000 \): \( 1000x = 523.523523...\).
3. Recognize that \( 523.523523... = 523 + x \).
4. Subtract \( x \) from both sides: \( 999x = 523 \).
5. Divide by \( 999 \): \( x = \frac{523}{999} \).

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