zoinnk wrote:

jugolo1 wrote:

I am looking for an approach to determine if a fraction has a terminating decimal or not without calculating it, for example:

which of the following has decimal equivalent to terminating decimal:

A- 10/189

B- 15/196

C- 16/225

D- 25/144

E- 39/128

the correct answer is E?

You need to prime factorize the denominator. If the denominator's prime factors are only 2 and/or 5, then the fraction is a terminating decimal

A: 189 = 7*3^4

B: 196 = 14^2 = 7^2*2^2

C: 225 = 15^2 = 3^2*5^2

D: 144 = 12^2 = 3^2*2^4

E: 128 = 2^7

I'd just add one thing here: you *must* make sure that your fraction is reduced before you apply this test. It's not important in the above question, because all of the fractions are already reduced, but it could be important in a different question. For example, you might see:

51/300

and if you just steam ahead and prime factorize that denominator, you'll get 3*2^2*5^2, which might make you think that it's not a terminating decimal. It is, however; when you reduce the fraction, the 3 in the denominator disappears:

51/300 = 17/100 = 0.17

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