The quotient of two positive integers is $\frac{5}{2}$ and their product is 160. What is the value of the larger of the two integers?
Let $2x$ be the smaller integer.  Then the larger integer is $5x$.  The product of the integers is 160, so $(2x)(5x)=160\implies 10x^2=160 \implies x^2=16$.  Since $x$ is positive, this implies $x=4$ which in turn implies that the larger integer is $5\cdot4=\boxed{20}$.