jmc

First, notice that each number in the numerator is a multiple of $2$, and each number in the denominator is a multiple of $3$. This suggests that each expression can be factored. Factoring gives: $\frac{2(1+2+3+\cdots +17)}{3(1+2+3+\cdots +17)}$ Since all the material in the parentheses is the same, the common factor in the numerator and the denominator may be cancelled, leaving $\frac{2}{3}$, which is option $\boxed{\frac{2}{3}}$.