Final answer:
Katrina is differentiating between terminating and repeating decimals through fraction division. Terminating decimals have a finite number of digits post-decimal, and repeating decimals have infinite repeating sequences. A fraction's denominator's prime factors determine if its decimal will terminate or repeat, and significant figures guide the precision of the result.
Step-by-step explanation:
Katrina is faced with dividing fractions to determine if they result in terminating decimals or repeating decimals. The process involves dividing the numerator by the denominator without the assistance of a calculator. In converting fractions to decimals, a terminating decimal is one that comes to a definite end, or 'terminates', after a certain number of decimals. A repeating decimal is one that has a sequence of digits after the decimal point that repeat indefinitely.
Without the specific fraction Katrina is working with, we cannot directly answer what results she got each time she subtracted, but we can elaborate on the method. When you divide by hand, you subtract the product of the digit you just found in your long division and the divisor from the current number. You then bring down the next digit and repeat the process. This subtraction continues until no digits are left or until a clear repeating pattern emerges.
The question of whether a decimal terminates or repeats is related to the properties of the fraction's denominator. A fraction will result in a terminating decimal if its denominator can be expressed as 2^n x 5^m, where n and m are non-negative integers (including zero), after it is fully simplified. This is due to the way our base-10 system works. In contrast, if the denominator has prime factors other than 2 or 5, it will result in a repeating decimal.
To confirm whether a decimal terminates or repeats and to determine the correct level of precision, understanding significant figures is essential. Calculators may provide a long string of digits, but we must use the number of significant figures in the original numbers involved to determine the actual precision of our answer. The guideline for significant figures varies between operations: one set of rules for addition and subtraction, and another for multiplication and division. In this process, it is crucial not to over-represent the precision of the answer and to maintain a proper handling of intermediate rounding.
The best practice is to carry forward all the digits through long division until you detect a repeating pattern or confirm that the decimal terminates (when no remainder occurs). Katrina should continue with her division until she detects a repeating pattern or confirms a terminating decimal, considering the rules of significant figures.