A vertical rectangular gate of 4m x 2m is hinged at a point 0.25 m below the centre of gravity of the gate. If the total depth of water is 7 m what horizontal force must be applied at the bottom to keep the gate closed?

As per the figure in your question....F is the hydrostatic force applied by water on the gate and P the force required to keep the gate closed. Here distance of P from hinge is 0.75m and distance of F from hinge is (0.78-0.75)m.

Take moment about hinge, then

P x 0.75 = F x 0.03

Here F = (density x g) x (Area of gate) x (distance of F from top surface of water)

       =(1000 x 9.81) x (4 x 2) x (5+1) = 470.88 KN

hence from moment equation we have P=(470.88 x 0.03)/(0.75) = 18.83 KN