A spaceship moves at a speed of $\frac{c}{4}$ away from the earth. As measured by a clock on the spaceship, the time interval between the maintenance check on the ships main computer is $15$ months. In the reference frame of Earth, what is the time interval between the maintenance checks on the ship's main computer?

The correct option is A Always less than $15$ months
We have, time dilation $T=T_0/\sqrt{1-v^2/c^2}$ ,

given $T=15months$ , $v=c/4$ ,

therefore , $15=T_0/\sqrt{1-\frac{c^2/4^2}{c^2}}$

or $T_0=\frac{15\times 15}{16}=14.0$ months

so on earth time interval is less than $15$ months because in spaceship (moving frame) clock will be seen to be running slow or dilated.