Let an annulus be ${1\leq x^2+y^2\leq4}$ and $I^2=[0,1]^2$.
The annulus is given the equivalence relation $(x,y)$~$(x',y')$ and $(x',y')$~$(2x,2y)$ if $x^2+y^2=1$
While $I^2$ is given the following equivalence relation: $(x,y)$~$(x',y')$ and $(0,y)$~$(1,y')$ and $(x,1)$~$(x',0)$ if $0\leq x,y \leq 1$
I tried to establish homeomorphisms between the quotient spaces and the torus but that method seems to be too long and tedious. Any suggestions?
Consider expressing the annulus as $I \times S^1$: e.g. via $\varphi: [1,2] \times S^1 \rightarrow A$ given by $\varphi(r, \theta) = (r cos \theta, r sin \theta)$. Then the relation on the annulus is given by identifying the endpoints of $[1,2]$, giving something homeomorphic to $S^1 \times S^1$.
That gives one homeomorphism to the torus. Given that you recognised that the torus was involved I believe you can build the other.