Which of the following operations has the same effect on a number as multiplying by $\dfrac{3}{4}$ and then dividing by $\dfrac{3}{5}$?
$\text{(A)}\ \text{dividing by }\dfrac{4}{3} \qquad \text{(B)}\ \text{dividing by }\dfrac{9}{20} \qquad \text{(C)}\ \text{multiplying by }\dfrac{9}{20} \qquad \text{(D)}\ \text{dividing by }\dfrac{5}{4} \qquad \text{(E)}\ \text{multiplying by }\dfrac{5}{4}$
Dividing by $\frac{3}{5}$ is the same as multiplying by its reciprocal, $\frac{5}{3}$.
Multiplying a number by two numbers is the same as multiplying the number by the product of those two numbers.  Thus, multiplying by $\frac{3}{4}$ and then $\frac{5}{3}$ is the same as multiplying by $\frac{3\cdot 5}{4\cdot 3} = \frac{5}{4}$, and the answer is $\boxed{E}$.