In simplest form, what type of decimal expansion will ( \frac{63}{231} ) have?

Correct answer: Non-terminating recurring

First reduce the fraction before deciding its decimal expansion. The numerator 63 and denominator 231 have a common factor of 21, so 63/231 becomes 3/11. For a fraction in simplest form, the decimal terminates only if its denominator has no prime factors other than 2 and 5. Since the denominator here is 11, the decimal cannot terminate.

Dividing 3 by 11 gives 0.272727 and the digits 27 continue repeating forever. Thus the decimal expansion is non-terminating recurring, not non-terminating non-recurring. The correct choice is option C. The reduction step is important because the denominator must be inspected only after all common factors have been cancelled; in this case, the simplified denominator still contains 11.