Replace each variable with a positive rational number (i.e. fraction) such that all of the following equations simultaneously hold true. Note that 0/n is not considered positive.

`
O + N + E = 1
T + W + O = 2
T + H + R + E + E = 3
F + O + U + R = 4
F + I + V + E = 5
S + I + X = 6
S + E + V + E + N = 7
E + I + G + H + T = 8
`

Additional constraints:

1) All fractions must have the same denominator, which must be as small as possible.

2) All variables must be distinct.

3) Reducible fractions, i.e. those in which the numerator and denominator share a common divisor greater than 1, are not allowed.

Hint: there are four distinct solutions. For extra credit, find the one with the lowest possible sum of the 14 variables.

This problem was inspired by a similar submission by pcbouhid (see http://perplexus.info/show.php?pid=3872)