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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)
4/7=0.571428571428571428571428...(prime number 7)
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)
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. 


I agree with Emily. 
Hello Bill and Emily,
                            Would you like to give your explanation seeing that you think my explanation is too confusing ? If you have a better, less confusing, explanation, I would sure like to see it.
Your answer is good for a mathematician but is not good for a struggling math student. This site is to help students not to impress math capable individuals. Short and to the point answers are always best. 
Bill,  I am a retired math teacher and have taught this same material to 7th and 8th grade students for over 33 years.
By the way, there is no other way to teach this material except by trial and error and that is not how you learn mathematics.
Arthur, with all due respect, I have nearly as much experience teaching as you and I did my master thesis on fractions but any teacher who says "there is no other way to teach" any math concept is not the teacher I would choose. I am very experienced but I continue to learn new teaching strategies for different math concepts. 
Bill, if you have another method to determine if the decimal equivalent of a given fraction is terminating or repeating please enlighten me.
My answer to the student's question is below. It's been there since 2013. I give answers with students in mind so mine is not elaborate. I can give you many ways to teach a student about terminating and non terminating rational numbers. Most students need it to be explained in a way that makes sense to them. That's what I always try to do. 
Bill, the student asked if there was a way to determine if the decimal equivalent is terminating or repeating. I gave the student all of the information he or she asked for. If the student didn't understand part of my explanation, I certainly would have explained that part to him or her again but there was no response after my post. I saw your answer back in 2013. If you have a fraction, how can you tell if it is going to be terminating or repeating without actually doing the division ? That is what the student was asking, and I explained that. What I disagree with basically is that you should not use trial and error to get the answer. Also, you don't always have to divide to get an answer, as I'm sure you know.
For example:  17/25=(17*4)/(25*4)=68/100=0.68. Also, what if the student asked how you could tell exactly how many digits will be in the repetend when you have a repeating decimal ? That's another whole explanation, which I also teach but left out of this explanation because it was not asked for by the student. For example, if you gave the student 5/7 and told the student to write it as a repeating decimal (he or she should already know that it is a repeating decimal from my explanation. lol), the student would have to divide 6 times and on the 7th division the student would see the repetend start to repeat. Repeating decimals are not as simple as they may appear to be. What if you gave the student 6667/30,000 and told the student to write it as a repeating decimal ? 6667/30,000=0.2222333333333.... I think the student might think that the "2" repeats when it's the "3" that repeats. I like to challenge my students, or at least I did when I was a teacher. Anyway, I think my explanation answered the question. Whether the student understood it or not is another question. This is why we give homework over and over again. As you know, that's how we learn, by practicing. It would be interesting to get the opinion of my explanation by other tutors on this website and also how other tutors teach this concept. One last comment. There will always be students who ask "why does this work" or "why does this happen". You have to be prepared to give a valid mathematical reason. There is a reason why 9/40 terminates and there is a reason why 3/7 repeats and you have to be able to answer the question. It's just like dividing with fractions. Why do you use the reciprocal of the divisor and change the sign to multiplication ? There is a mathematical reason for this. Respond back if you like. Have a nice day.
Arthur D.
You're welcome, Mike, and thank you for the kind words.
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.