I want to deduce Length Contraction using Time Dilation(which has already been deduced), but I encountered a problem that I feel tough.

I first assume two observers, $A$ and $B$ are in two frames with relative speed $v$ between them, and assume just next to B, there is a stick which B measures to be of length $l$.

Now I let $B$ record the time needed for $A$ to traverse the stick, thus $t_B={l\over v}$, note that $t_B$ is the time measured in $B$'s frame using $B$'s clock.

Then I go to $A$'s frame, and I assert that $A$ will record a time $t'_A={l'\over v}$ for $A$ to traverse the stick.

Now the only thing left is to relate $t'_A$, the time measured in $A$'s frame using $A$'s clock, and $t_B$.

So my questions are:

1). How is it justified that the relative speed between the two frame is the same as measured by each observer in each frame?

2). How to use time dilation formula to relate $t'_A$ with $t_B$?

Please help me to clarify my misconception: Shall I let $B$ look at $A$'s clock in $B$'s frame, or shall I let $A$ to look at $B$'s clock in $A$'s frame?