a gate 6m wide requires a force of 5.2 kg.wt. applied at the end to open it. what force would have to be applied (A) at the middle and (B) at 1.5m from the hinges, to open it?

the product of force and lever arm must be constant

F * x = 5.2 * g * 6 = 31.2 g

so A
F * 3 = 31.2 g
F = 10.4 g

B
F * 1.5 = 31.2 g
F = 20.8 g

The g is because I assume your mass is hanging from a pulley pulling on the gate. Kg is mass, not force.