Question

a 100 g box is sliding down an inclined plane. which box will move with the greatest speed as a result of the applied force?

Explanation:

Step1: Recall Newton's second law

$F = ma$, where $F$ is net - force, $m$ is mass and $a$ is acceleration. The greater the acceleration, the greater the speed change over time (assuming starting from rest or with the same initial speed). The mass of the box is constant ($m = 100g=0.1kg$).

Step2: Analyze the direction and magnitude of the applied force

The force acting down - the - incline will increase the acceleration of the box sliding down the incline. A force acting up - the - incline will decrease the acceleration.
The first two cases have forces acting down the incline. The first has a force of $F_1 = 10N$ and the second has a force of $F_2=20N$. The second two cases have forces acting up the incline.
Since $F = ma$, for a constant mass, a larger force in the direction of motion (down the incline here) will result in a larger acceleration.

Answer:

The box with a $20N$ force acting down the incline will move with the greatest speed.