Four forces are acting at a point P in equilibrium as shown in figure. The ratio of force F₁ to F₂ is 1:x where x= __________.

Idea: equilibrium means the components balance in each direction separately. Take the vertical as the reference, since both slanting forces sit at 45° to it.

Measuring x to the right and y upward, and using sin 45°=cos 45°=1/(√2):

the 2 N force, up and to the left, is (-√2, +√2);

the 1 N force, up and to the right, is (+1/(√2), +1/(√2));

F₁ is (+F₁, 0) and F₂ is (0, -F₂).

Horizontal: -√2+1/(√2)+F₁=0, so

F₁=√2-1/(√2)=(2-1)/(√2)=1/(√2).

Vertical: √2+1/(√2)-F₂=0, so

F₂=(2+1)/(√2)=3/(√2).

(F₁)/(F₂)=(1/√2)/(3/√2)=1/3, so x=3.

The √2 cancels in the ratio, which is why the answer is a whole number.