Total and strict transforms of blowups

Question

I have a rather simple question about the nature of blowups: Let's say I have $\pi: X \rightarrow Y$, which is the blowup of a center $Z$ in $Y$. Denote the exceptional divisor by $E$. Is it correct, that I can write $\pi^{*}(V) = \widetilde{V} + mE$ for any $V$ in $Y$, where $m$ is the multiplicity and $\widetilde{V}$ is the strict transform? Are there any cases, where I might not get an equality here? I have seen this equation being used in books, but I am not quite sure if it is only true in some special cases - what is the reasoning behind this?