Arthur D. answered • 10/04/16
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How to determine if the decimal expansion of a fraction is repeating or terminating:
Make sure that the fraction is in lowest terms.
Factor the denominator as a product of prime factors.
If the prime factorization contains only 2s, only 5s, or a combination of 2s and 5s only, then the decimal expansion will always be terminating. If the prime factorization of the denominator contains any other prime numbers, like a 3, a 7, an 11, and so on, then the decimal expansion will always be repeating.
Examples:
The following fractions all have decimal expansions that are terminating:
1/2, 3/4, 4/5, 7/8, 3/10, 15/16, 17/20, 23/25, 21/32, 13/40, 47/50, 45/64, 77/80, 87/100, 123/125,...
Here's why...
2=2
4=2*2
5=5
8=2*2*2
10=2*5
16=2*2*2*2
20=2*2*5
25=5*5
32=2*2*2*2*2
40=2*2*2*5
50=2*5*5
64=2*2*2*2*2*2
80=2*2*2*2*5
100=2*2*5*5
125=5*5*5
1/2=0.5
3/4=0.75
4/5=0.8
7/8=0.875
3/10=0.3
15/16=0.9375
17/20=0.85
23/25=0.92
21/32=0.65625
13/40=0.325
47/50=0.94
45/64=0.703125
77/80=0.9625
87/100=0.87
123/125=0.984
The following fractions all have decimal expansions that are repeating:
2/3, 5/6, 3/7, 8/9, 7/11, 5/12, 12/13, 11/14, 14/15, 15/17, 13/18, 18/19, 19/21, 19/22, 17/27, 31/37,...
2/3=0.666...
5/6=0.8333...
3/7=0.428571428571428571...
8/9=0.888...
7/11=0.636363...
5/12=0.41666...
12/13=0.923076923076923076...
11/14=0.7857142857142857142...
14/15=0.9333...
15/17=you can do this one if you like
13/18=0.7222...
18/19=you can do it
19/21=0.904761904761904761...
19/22=0.8636363...
17/27=0.629629629...
31/37=0.837837837...
