Is Nora or Sofia correct? Explain your reasoning.

  Nora is correct

Step-by-step explanation:

Sophia used an appropriate method for computing the value of the repeating decimal fraction ...

  0.1212_12

Close examination of the given decimal fraction shows that it is non-repeating, so is an irrational number. It cannot be represented by the ratio of integers.

Nora is correct in saying Sophia made a mistake.

___

Sophia depended on 100x -x cancelling some part of the fraction, so the result was a finite decimal. For repeating 2-digit fractions, this works well:

However, the decimal does not repeat, so this does not work:

  100x = 12.1121112

  100x -x = 12.1121112 -0.121121112 = 11.99099009900099... non-repeating

That is, the decimal part is not cancelled by Sophia's work, as we said above.