Birds and cages

If the number of cages is x (x-1)×13=8x+12 by solving equation x=5 Number of birds (5-1)×13=52

Let there are \color\red{x} number of birds and \color\pink{y} number of cages.. Now we can form two equations for the two cases mentioned in the question... $x - 13(y-1)= 0………(1)$ $x - 8y= 12……………(2)$

Applying $(1)-(2)$ ,,,

$5y = 25$ $\Rightarrow y= 5$ .... Plugging $y=5$ in equation $(1)$ we get, $x- 13*5 = -13$ \Rightarrow \boxed{\color\red{x= 52}} .... So, there are $52$ birds and $5$ cages..