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Put the steps in order for changing the repeating decimal, which is rational, to a ratio or fraction. 0.523523... = what fraction?

Answer Choices that you have to order:
Subtract 1x=0.523523
1000x= 523.523523
Let x= 0.523523
999x=523
x=523/999
now put these in order

1 answer

To convert the repeating decimal 0.523523... into a fraction, the steps should be ordered as follows:

  1. Let \( x = 0.523523... \)
  2. Multiply by 1000 to shift the decimal point: \( 1000x = 523.523523... \)
  3. Subtract the original equation from this new equation: \( 1000x - x = 523.523523... - 0.523523... \)
  4. This simplifies to \( 999x = 523 \)
  5. Solve for \( x \): \( x = \frac{523}{999} \)

So the correct order of the steps is:

  1. Let \( x = 0.523523... \)
  2. \( 1000x = 523.523523... \)
  3. \( 999x = 523 \)
  4. \( x = \frac{523}{999} \)

Note that the step "Subtract 1x=0.523523" is part of the subtraction process in step 3 and can be omitted from the ordered list since it is already included in the equation.