Sets $A$ and $B$, shown in the Venn diagram, are such that the total number of elements in set $A$ is twice the total number of elements in set $B$. Altogether, there are $9$ elements in the union of $A$ and $B$, and their intersection has $3$ elements. What is the total number of elements in set $A$?
Let $A$ have $x$ elements and $B$ have $y$ elements. We are given that $x = 2y$. Also, we are given that $x + y - 3 = 9$, or $x + y = 12$.
Substituting $x = 2y$ into $x + y = 12$, we get $3y = 12$, so $y = 4$. Then $x = 2y = 8$.
Therefore, the total number of elements in set $A$ is $\boxed{8}$.
Let's assume that the total number of elements in set $B$ is $x$.
According to the information given, the total number of elements in set $A$ is twice the total number of elements in set $B$. Therefore, the total number of elements in set $A$ is $2x$.
The number of elements in the union of $A$ and $B$ is $9$, which means $A \cup B = 9$.
Since the intersection of $A$ and $B$ has $3$ elements, we can write $A \cap B = 3$.
From the Venn diagram, we can see that $A = (A \cup B) - (A \cap B)$. So, we substitute the values that we know:
$A = 9 - 3 = 6$
Therefore, the total number of elements in set $A$ is $6$.