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How can we tell if a given fraction will terminate or recur as a decimal fraction?

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2 Answers

the first thing to do is write the fraction in simplest form
next, write the prime factorization of the denominator
if the prime factorization of the denominator 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 be terminating; also if the denominator is a power of ten to begin with,  you don't have to factor, the decimal expansion will be terminating
for example, 3/10=0.3, 37/100=0.37, 528/1000=0.528, 345/10,000=0.0345, and so on
let's look at other examples:
1/2=0.5 because the denominator is 2
3/4=0.75 because 4=2x2
2/5=0.4 because the denominator is 5
7/8=0.875 because 8=2x2x2
13/16=0.8125 because 16=2x2x2x2
19/20=0.95 because 20=2x2x5
21/25=0.84 because 25=5x5
29/32=0.90625 because 32=2x2x2x2x2
37/40=0.925 because 40=2x2x2x5
49/50=0.98 because 50=2x5x5
59/64=0.921875 because 64=2x2x2x2x2x2
27/80=0.3375 because 80=2x2x2x2x5
119/125=0.952 because 125=5x5x5
do you see the pattern ? all 2's, all 5's, or a combination of 2's and 5's only
now for repeating decimals
don't forget to simplify the fraction first
write the prime factorization of the denominator
if the denominator is a prime number other than a 2 or a 5, or if the prime factorization of the denominator contains any prime numbers other than 2's or 5's, the decimal expansion of the fraction will be repeating
let's look at some examples:
2/3=0.666...(the prime number 3 makes the decimal repeat)
5/6=0.8333...(6=2x3)
4/7=0.571428571428571428571428...(prime number 7)
5/9=0.555...(9=3x3)
7/11=0.636363...(prime number 11)
5/12=0.41666...(12=2x2x3)(the two 2's give us the 41 called the lag and the 3 gives us the 666...)
13/15=0.8666...(15=3x5)(the 5 gives us the lag which is 8 and the 3 makes the decimal then repeat itself)
19/24=0.791666...(24=2x2x2x3)
26/27=0.962962962...(27=3x3x3)
24/37=0.648648648...(prime number 37)
I hope you understand from these examples.
Keep in mind that the decimal expansion of certain fractions will contain the maximum number of digits in the repetend(what repeats is called the repetend). The maximum number of digits that can possibly repeat is 1 less than the denominator. For example, look at 4/7 above;six digits repeat, which is 1 less than 7 !! Another example of a fraction yielding the maximum number of digits is any fraction whose denominator is 19; 3/19, 7/19, 14/19, 18/19. If you want to try one of these to see the 18-digit repetend, by all means try it. 
 
 
 
 
 
Hi Henriettas,
 
This is a really good question.  The best answer I can give is take the top (numerator) and divide it by the bottom (denominator). 
 
There are certain fractions that do not terminate like 1/3, 1/9, 1/7.  Also, anytime there is a fraction with a 9, 99, 999, etc. in the denominator it will be a repeating, non-terminating decimal.
 
So, try these:
 
1/5 =
 
2/3 =
 
3/8 =
 
4/7 =
 
24/99 =
 
 
Now, notice that the ones that terminate can be changed to tenths, hundredths, thousandths, etc.  The ones that do not terminate cannot be changed to tenths, hundredths, thousandths, etc.
 
Bill