There are some cows, bulls, and 45 hens in a group. One caretaker...

Let the number& of cows =~$x$

So, the number of bulls =~$2x$

Number of hens $=45$
Number of caretakers required =~$\dfrac{x + 2x + 45}{15}$
Number of feet =

(Number~of~feet~of& cow) $\times$ (Number~of~cows) $+$~(Number~of~feet& of& bull $\times$ (Number~of~bulls) $+$(Number~of~feet~of& hen) $\times$~(Number~of~hens)& $+$(Number~of~feet~of& caretaker) $\times$ (Number~of~caretakers)~

=~$4x + 8x + 90 + 2 ( \dfrac{x + 2x + 45}{15} )$
Number of head =~(Number~of~head& of& cow) $\times$ (Number~of~cows) $+$(Number~of~head~of& bull $\times$~(Number~of~bulls) $+$(Number~of~head~of& hen) $\times$ (Number~of~hens) $+$(Number~of~head~of& caretaker) $\times$ (Number~of~caretakers)~

=~$x + 2x + 45 + \dfrac{x + 2x + 45}{15}$

Since, $4x + 8x + 90 + 2\left(\dfrac{x + 2x + 45}{15}\right) - \left(x + 2x + 45 + \dfrac{x + 2x + 45}{15}\right) = 186$
~ ~$9x + 45 + \dfrac{x + 2x + 45}{15} = 186$

$9x + \dfrac{x + 2x + 45}{15} = 141$

$138x = 2115 - 45$

$x = \dfrac{2070}{138} = 15$
~ ~ ~ &
Number of caretakers $=\dfrac{x + 2x + 45}{15} = \dfrac{15 + 2 \times 15 + 45}{15} = \dfrac{90}{15} = 6$

Hence, the number of caretakers is $6$.