Rabbits and cages

$x=number~of~rabbits$

$y=number~of~cages$

$\dfrac{x}{y-1}=13$ $\implies$ $\boxed{1}$

$\dfrac{x}{y}=8+\dfrac{12}{y}$ $\implies$ $\boxed{2}$

Solving for $x$ in terms of $y$ in $\boxed{1}$ then substituting to $\boxed{2}$ , we have

$\dfrac{13y-13}{y}=8+\dfrac{12}{y}$ $\implies$ $\boxed{3}$

Multiplying $\boxed{3}$ by $y$ to remove denominators, we have

$13y-13=8y+12$

It follows that, $y=5$ .

Finally,

$x=13(5)-13=$ $\color{#D61F06}\boxed{52}$

If I put 13 rabbits in each cage, one cage is left empty.

so, the answer must be divisible by 13 and the answer is 52.........[the only option divisible by 13]

now,making it more sure.if after putting 13 rabbits in 1 cage,there still remains 1 cage,so there is 5 cages.

and putting 8 rabbits in 1 cage,there will be 40 rabbits in 5 cages.and remains 12 rabbits not caged.[ $(52-40=12)$

so, there are 52 rabbits.

Since we know that if 13 bunnies are in each cage and one is empty, the answer must be divisible by 13. If we look on the answer choices, only A and D fit. Next we find that if 8 bunnies are put in each cage, 12 rabbits are uncaged. A does not work because it is too small. Therefore, using process of elimination, we find that D is the correct answer.

x = number of cages

As, 13 rabbits are put in each cage and one cage is left empty,

Number of rabbits = 13(x-1) =13x-13

As, 8 rabbits are put in each cage and 12 rabbits are left uncaged

Number of rabbits = 8x+12

So, 13x-13 = 8x+12

or, 13x-8x = 12+13

or, 5x = 25

or x = 5

Finnaly,

Number of cages = 5

Number of rabbits = 13*5 - 13

                            = 52

In another method,

Number of cages = 5

Number of rabbits = 8*5+12

                            = 52

Let take number of cage =y then rabbit =(y-1)×13 again y×8+12 no of rabbits equal then (y-1)×13=y×8+12.by soving it give 52

Two conditions need to be fulfilled.

One: the number of rabbits must be greater than 20 (Second statement indicates so)

Two: the number of rabbits must be divisible by 13 (First statement indicates so)

Therefore only OPTION D (52) satisfies.

Because there are 13 rabbits put in each cage with none left over, you know that the answer is some multiple of 13. This narrows it down to 13 and 52. It cannot be 13 because 8 + 12 (the minimum amount required to make the second statement true) is more than 13. So the answer is 52