jmc

We multiply both sides of the equation by $10x$ to clear the fractions, leaving us with $2x^2+2=5x$. Rearranging the terms, we have $2x^2-5x+2=0$. We can now solve for $x$ by factoring: $(2x-1)(x-2)=0$. We could also use the quadratic formula: $$x=\frac{5\pm \sqrt{(-5{)}^2-4(2)(2)}}{4}.$$Either way, we find that $x=1/2$ or $x=2$. Since we want the largest value of $x$, our answer is $\boxed{2}$.