Analysis

This problem examines the method to find the fifth term of a geometric sequence by using the general formula of a geometric sequence to find the first term and the common ratio, from which the value of $a_5$ can be determined.

Solution

Given that the geometric sequence $\{a_n\}$ satisfies $a_1+a_3=10$ and $a_2+a_4=5$,
we have $\begin{cases}a_1+a_1q^2=10 \\ a_1q+a_1q^3=5\end{cases}$,
from which we find $a_1=8$ and $q=\dfrac{1}{2}$,
thus $a_5=a_1q^4=8×\dfrac{1}{16} =\dfrac{1}{2}$.
Therefore, the correct choice is $\boxed{B}$.