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

Question

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