How can you tell whether or not a rational expression written in fractional form will terminate or repeat?

Question

I would like to know how to tell if it terminates or repeats  without finding the decimal expression

Answer 1

First of all the fraction must be in simplest form.

Factor the denominator as a product of prime factors.

If the prime factorization contains all 2's, all 5's, or a combination of 2's and 5's only, then the decimal expansion of the fraction will terminate.

If there are any other prime numbers in the prime factorization of the denominator that are not either 2's or 5's, then the decimal expansion will be repeating.

Here are examples of fractions that will terminate:

1/2=0.5

3/4=0.75 because 4=2x2

4/5=0.8

3/8=0.375 because 8=2x2x2

7/10=0.7 because 10=2x5

9/16=0.5625 because 16=2x2x2x2

13/20=0.65 because 20=2x2x5

19/25=0.76 because 25=5x5

37/40=0.925 because 40=2x2x2x5

49/50=0.98 because 50=2x5x5

51/64=0.796875 because 64=2x2x2x2x2x2

63/80=0.7875 because 80=2x2x2x2x5

97/100=0.97 because 100=2x2x5x5

58/125=0.464 because 125=5x5x5

other denominators that will yield terminating decimals are 128, 160, 200, 250, 320, 500 and many, many, more.

 

Here are some fractions that will repeat:

2/3=0.666...

5/6=0.8333... because 6=2x3

5/7=0.714285714285...

8/9=0.888... because 9=3x3

9/11=0.818181...

11/12=0.91666... because 12=2x2x3

13/15=0.8666... because 15=3x5

17/18=0.9444... because 18=2x3x3

15/22=0.6818181...because 22=2x11

17/27=0.629629629... because 27=3x3x3

32/37=0.864864864...

19/44=0.43181818... because 44=2x2x11

71/72=0.986111... because 72=2x2x2x3x3

and there are many, many more denominators that will yield a repeating decimal

Notice that in the prime factorization of the denominator whose fraction yields a repeating decimal, if there are any 2's or any 5's there will be a lag that precedes the repetend.

example: 15/22=0.6818181... where 6 is the lag and 81 is the repetend.

62/75=0.82666... where 82 is the lag and 6 is the repetend.