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A and C are out because the denominators aren't even.

Left with BDE

D is 5^2 / 12^2, since 5/12 isn't a terminating decimal, I suppose the square of them aren't too.

D is out

It's E, because the denominator is a multiple of 2. (But I'm not sure, because my approach is rather unsustained. However, I guess there is no number which, if divided by 2 or a multiple of it, yields an non-terminating decimal.)

( 196 is the square of the odd number 3; I think the division by an odd number ( I used this approach also above) yields an non-terminating decimal.)

E.
Lets look at each fraction
a. 10/189=10*1/(3^3)*1/7 each of this simpe fractions does not lead to derminating decimal
b.15/195=15*1/13*1/13 the same thing
c. 16/225 = 16*1/5*1/5*1/9 no
d. 25/144=25*1/(3^2)*1/(2^4) 1/3 is present and does not give determinating decimal
e. 39/128=39/(2^7) Yes. 1/2 =0.5 and (0.5)^7 has the end.

I couldn’t help myself but stay impressed. young leader who can now basically speak Chinese and handle things alone (I’m Korean Canadian by the way, so...