Step 1) Subtract #color(red)(P)# from each side of the equation to isolate the #r# term while keeping the equation balanced:

#A - color(red)(P) = -color(red)(P) + P + Prt#

#A - P = 0 + Prt#

#A - P = Prt#

Now, divide each side of the equation by #color(red)(P)color(blue)(t)# to solve for #r# while keeping the equation balanced:

#(A - P)/(color(red)(P)color(blue)(t)) = (Prt)/(color(red)(P)color(blue)(t))#

#(A - P)/(Pt) = (color(red)(cancel(color(black)(P)))rcolor(blue)(cancel(color(black)(t))))/(cancel(color(red)(P))cancel(color(blue)(t)))#

#(A - P)/(Pt) = r#

#r = (A - P)/(Pt)#

Or

#r = A/(Pt) - P/(Pt)#

#r = A/(Pt) - color(red)(cancel(color(black)(P)))/(color(red)(cancel(color(black)(P)))t)#

#r = A/(Pt) - 1/t#