Let the cost price per kg of type $A,$ and $B$ be $\text{Rs.} x$ and $\text{Rs.} y$ respectively.
When, $A$ and $B$ are mixed in the ratio of $3:2,$ then the profit is $10\%,$
So, selling price $:$
$ \left( \frac{3x+2y}{5} \right) \times \frac{110}{100} = 40 $
$ \Rightarrow \left( \frac{3x+2y}{5} \right) \times \frac{11}{10} = 40 \quad \longrightarrow (1) $
When, $A$ and $B$ are mixed in the ratio of $2:3,$ then the profit is $5\%.$
So, selling price $:$
$ \left( \frac{2x+3y}{5} \right) \times \frac{105}{100} = 40 $
$ \Rightarrow \left( \frac{2x+3y}{5} \right) \times \frac{21}{20} = 40 \quad \longrightarrow (2) $
On equalling equation $(1),$ and $(2),$ we get
$ \left( \frac{3x+2y}{5} \right) \times \frac{11}{10} = \left( \frac{2x+3y}{5} \right) \times \frac{21}{20} $
$ \Rightarrow (3x+2y) \times 22 = (2x+3y) \times 21 $
$ \Rightarrow 66x + 44y = 42x + 63y $
$ \Rightarrow 24x = 19y $
$ \Rightarrow \boxed{\frac{x}{y} = \frac{19}{24}} $
$\therefore$ The cost price per kg of $A$ and $B$ is $19 : 24.$
Correct Answer $: \text{B}$