functions – Is there a way to use identities in Mathematica efficiently?

Imagine I have defined a variable
l=p/(a*b)
Let’s say I perform some operations, and I get the final answer to be p/a.
But all I really care about is how this expression looks in terms of l and b. Is there a way to use the relation l=p/(a*b) to convert p/a to l/b?

Of course this is a very trivial example, it gets trickier when you have other, more complicated relations. I have tried the replace command /.{p/(a*b) -> l} but it does not work.

I would appreciate any advice you have!