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 $A$?

Answer 1

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 $\boxed{8}$.