Answer
The particle of mass $3m$ must be placed between the two particles but closer to the less massive particle.
Work Step by Step
We can find the magnitude of the gravitational force on the particle of mass $3m$ due to the particle of mass $m$:
$F = G~\frac{(m)(3m)}{r_1^2} = \frac{3Gm^2}{r_1^2}$
Note that this force is directed toward the particle of mass $m$.
We can find the magnitude of the gravitational force on the particle of mass $3m$ due to the particle of mass $2m$:
$F = G~\frac{(2m)(3m)}{r_2^2} = \frac{6Gm^2}{r_2^2}$
Note that this force is directed toward the particle of mass $2m$.
For the net gravitational force on the particle of mass $3m$ to be 0, the two gravitational forces must point in opposite directions. Therefore, the particle of mass $3m$ must be placed between the two other particles.
For the net gravitational force on the particle of mass $3m$ to be 0, the two opposing gravitational forces must have the same magnitude. Therefore, the particle of mass $3m$ must be placed closer to the particle of mass $m$ such that $r_1 \lt r_2$
The particle of mass $3m$ must be placed between the two particles but closer to the less massive particle.
