2001 AMC 10 Problems/Problem 10

Problem

If $x$ , $y$ , and $z$ are positive with $xy = 24$ , $xz = 48$ , and $yz = 72$ , then $x + y + z$ is

$\textbf{(A) }18\qquad\textbf{(B) }19\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\textbf{(E) }24$

Look at the first two equations in the problem.

$xy=24$ and $xz=48$ .

We can say that $2y=z$ .

Given $2y=z$ , we can substitute $z$ for $2y$ and find

$2y^2=72$ $y^2=36$ $y=6$ $2y=z=12$ .

We can replace y into the first equation. $6x=24$ $x=4$ .

Since we know every variable's value, we can substitute it in for $x+y+z = 4+6+12 = \boxed{\textbf{(D) }22}$ .