Answer: 36

Using a diagram

If one third of the flock are goats, then two thirds must be sheep. 

Image

 

There are 12 more sheep than goats:

Image

 

So altogether there are 12 + 12 + 12 = 36 animals.

Image

 

Using fractions and words

If one third of the flock are goats, then two thirds must be sheep.

If there are 12 more sheep than goats: difference between one third and two thirds of the flock must be 12 animals.

So one third of the flock is 12 animals. So the whole flock is 12 $\times$ 3 = 36 animals.

Using proportion and words

If one third of the flock are goats, then two thirds must be sheep, so there must be twice as many sheep as goats.

There are 12 more sheep than goats, so doubling the number of goats is the same as adding 12. So there must be 12 goats.

That means there must be 24 sheep, so there are 36 animals altogether.

Using fractions and algebra

Suppose there are $A$ animals in the flock. Then there are $\frac{1}{3}A$ goats in the flock, so there must be $\frac{2}{3}A$ sheep.

There are 12 more sheep than goats, so $\frac{1}{3}A+12=\frac{2}{3}A\Rightarrow12=\frac{2}{3}A-\frac{1}{3}A=\frac{1}{3}A$.

So $\frac{1}{3}A=12\Rightarrow A=36$.

Using proportion and algebra

If one third of the flock are goats, then two thirds must be sheep, so there must be twice as many sheep as goats. So $s=2g$, where $g$ represents the number of goats in the flock and $s$ represents the number of sheep in the flock.

There are $12$ more sheep than goats, so $s=g+12$.

But if $s=2g$ and $s=g+12$, then $2g=g+12\Rightarrow g=12$.

Since a third of the flock are goats, there are $3g=3\times12=36$ animals in the flock.