Put the steps in order for changing the repeating decimal, which is rational, to a ratio or fraction.
0.523523... = what fraction
1. Write down the given repeating decimal as an equation: let x be the repeating decimal, so x = 0.523523...
2. Multiply both sides of the equation by a power of 10 to eliminate the repeating part. In this case, since there are three digits that repeat, multiply by 1000 (10^3).
1000x = 523.523523...
3. Subtract the original equation (step 1) from the equation obtained in step 2 to eliminate the repeating part.
1000x - x = 523.523523... - 0.523523...
999x = 523
4. Solve for x by dividing both sides of the equation by 999.
x = 523/999
5. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (gcd). In this case, gcd(523, 999) = 1, so the fraction cannot be simplified further.
x = 523/999
Thus, the fraction equivalent to the repeating decimal 0.523523... is 523/999.