From this moment, I hate animals! `Farmer's Brain`

Cows = ${x}$ , Bulls = ${y}$ , Hens = ${z}$ = 45, Caretakers or stockman ${c}$

${x}=$ $\frac{1}{2}$ ${y}$ => ${y}= 2{x}$

1 caretaker can take care of 15 animals, ie

1 caretaker = 15 animals

$\frac{1}{15}$ caretakers = 1 animal

Therefore, total no of animals ( ${x}$ + ${y}$ + ${45}$ ) = $\frac{1}{15}$ ( ${x}$ + ${y}$ + ${45}$ ) caretakers

= $\frac{1}{15}$ ( ${x}$ + ${2x}$ + ${45}$ ) caretakers = $\frac{1}{15}$ ( ${3x}$ + ${45}$ ) caretakers = $\frac{3}{15}$ ( ${x}$ + ${15}$ ) caretakers = $\frac{1}{5}$ ( ${x}$ + ${15}$ ) caretakers

So, total no of caretakers = $\frac{1}{5}$ ( ${x}$ + ${15}$ ) ---- $\boxed{1}$

Total no of heads is less than total no of legs by 186

ie, Total no of legs - Total no of heads = 186

Total no of heads = total no of animals + total no of caretakers ( no human or animals have 2 or more heads O-o) = ( ${3x}$ + ${45}$ ) + ( $\frac{1}{5}$ ( ${x}$ + ${15}$ )) = ( $\frac{1}{5}$ ( ${16x}$ + ${240}$ ))

Total no of feet/legs = 4 ( cows + bulls) + 2 ( hen + caretakers) = ( $4 \times {3x}$ ) + $2$ ( $45$ + $\frac{1}{5}$ ( ${x}$ + ${15}$ )) = $12{x}$ + $2(\frac{240 + {x}}{5})$ = $\frac{62{x} + 480}{5}$

Therefore, ( $\frac{62{x} + 480}{5}$ ) - (( $\frac{1}{5}$ ( ${16x}$ + ${240}$ ))) = 186

=> $\frac{46{x} + 240}{5}$ = 186

=> $46{x} + 240$ = 930

=> $46{x}$ = 690

=> ${x}$ = 15

Total no of cows = 15.

Total no of caretakers = $\frac{1}{5}$ ( ${x}$ + ${15}$ ) $\boxed{1}$

=> Caretakers = $\frac{1}{5}$ ( ${15}$ + ${15}$ ) = 6

Therefore, Total no of caretakers = 6 !!

Let the number of stockman be $w$

It is given that $x=\dfrac{1}{2}y \implies y=2x\implies$ Eq.(1)

We know that each stockman takes care of $15$ animals. Therefore,

$\dfrac{x+y+z}{15}=w$

Substitute $z=45$ and Eq.(1):

$x+2x+45=15w\\ 3x=15w-45$

$x=5w-15 \implies$ Eq.(2)

Given that:

Total legs $-$ Total heads $=186$

$(2w+4x+4y+2z) - (w+x+y+z) = 186\\ w+3x+3y+z=186$

Substitute $z=45$ , Eq.(1) and Eq.(2):

$w+3(5w-15)+3(2x)+45=186\\ w+15w-45+6(5w-15)+45=186\\ 16w+30w-90=186\\ 46w=276\\ w=\boxed{6}$