jmc

We take a common denominator: $$\frac{15}{11},\frac{19}{15},\frac{17}{13}=\frac{15\cdot 15\cdot 13}{11\cdot 15\cdot 13},\frac{19\cdot 11\cdot 13}{15\cdot 11\cdot 13},\frac{17\cdot 11\cdot 15}{13\cdot 11\cdot 15}=\frac{2925}{2145},\frac{2717}{2145},\frac{2805}{2145}.$$ Since $2717<2805<2925$ it follows that the answer is $\boxed{\text{(E)}\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}$. Another approach to this problem is using the properties of one fraction being greater than another, also known as the butterfly method. That is, if $\frac{a}{b}>\frac{c}{d}$, then it must be true that $a\ast d$ is greater than $b\ast c$. Using this approach, we can check for at least two distinct pairs of fractions and find out the greater one of those two, logically giving us the expected answer of $\boxed{\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}$.