Which of the following rational numbers can be expressed as a repeating decimal: 7/8 , 5/12 , 17/25 , or 31/40 ?

Answer 1

We can rewrite each fraction with a denominator in the form $2^a5^b$:

$7/8 = 7/(2^3) = 7\cdot5^{-3}$
$5/12 = (5\cdot2^1)/(2^2\cdot3) = 5\cdot2^{-1}\cdot3^{-1}$
$17/25 = 17/(5^2) = 17\cdot2^{-2}\cdot5^{-2}$
$31/40 = (31\cdot2^1)/(2^3\cdot5) = 31\cdot2^{-2}\cdot5^{-1}$

The only denominator that contains no factors of numbers other than 2 and 5 is $2^3\cdot5^2 = 40$. Thus 17/25 cannot be expressed as a repeating decimal, but the other three can. $\boxed{5/12 \quad 31/40 \quad 7/8}$