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?