Motivation behind word quotient Why set of all cosets of a subspace W of a vector space V is called quotient space. What is the motivation behind word quotient?
 A: Because the quotient of $V \times W$ by $W$ is $V$.
(more precisely, the quotient of $V \times W$ by the subspace $\{ 0 \} \times W$ is naturally isomorphic to $V$)
There are more general situations, but that it covers this particular situation is, in my opinion, a pretty strong motivation.
The idea is even clearer in the case of abelian groups; finite abelian groups are sort of like numbers. If $C_k$ denotes a cyclic group of $k$ elements, then the quotient of $C_{mn}$ by the subgroup (isomorphic to) $C_m$ is (isomorphic to) $C_n$.

The notion of quotient can even be applied to pure sets; if $\sim$ is an equivalence relation on a set $S$, then there is a quotient set $S/\!\!\sim$; typically we define this to be the set of equivalence classes.
If every equivalence class has exactly $m$ elements, then $|S /\!\!\sim\!\!| = \frac{|S|}{m}$
A: Two possible motivations:
For finite-dimensional vector spaces over finite fields,
$$
|V/W|=|V|/|W|\;.
$$
And for $m\mid n$,
$$
n\mathbb Z/m\mathbb Z\sim(n/m)\mathbb Z\;.
$$
More generally, see also the Wikipedia article on equivalence classes. Whenever the equivalence classes of an equivalence relation all have the same size $d$, forming the quotient with respect to that equivalence relation divides the size of the structure by $d$.
