In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $15$. What is the greatest possible perimeter of the triangle? 
$\textbf{(A) } 43\qquad \textbf{(B) } 44\qquad \textbf{(C) } 45\qquad \textbf{(D) } 46\qquad \textbf{(E) } 47$
Let $x$ be the length of the first side. 
The lengths of the sides are: $x$, $3x$, and $15$. 
By the [Triangle Inequality](https://artofproblemsolving.com/wiki/index.php/Triangle_Inequality),
$3x < x + 15$
$2x < 15$
$x < \frac{15}{2}$
The greatest integer satisfying this inequality is $7$.
So the greatest possible perimeter is $7 + 3\cdot7 + 15 =\boxed{\textbf{(A) } 43}$