Hydraulic lift connected to two pistons

Question

I just finished a problem, and got the correct answer, however I am confused on one of the mathematical operations that must be performed in order to get the correct answer.

The problem is as reads: We have a piston connecting two cylinders A and B of diameters 100mm and 25mm, respectively. The pressure acting on the outside of the entire piston is 100kPa, and the mass of the piston is 25kg, note that there is specific gravity present. A hydraulic lift pump at Cylinder A is pumped up to 500kPa. What is the gas pressure in cylinder B ?
So the formula for the answer is :
PressureB = [(PressureA × AreaA) - (MassPiston × 9.81) - (PressureOutside × (AreaA - AreaB ))] / AreaB
This all makes sense to me except for the difference in AreaA - AreaB multiplied by external Pressure. Why wouldn't we add the two areas instead to find the net force that the external pressure exerts ?
Any explanation will be appreciated.

NMech · Accepted Answer

you take the difference because the force due to pressure acting on the top surface has direction upwards, while the force on the bottom surface has the opposite direction (same direction as gravity).
So the two forces cancel each other. Becauce the $F_{top}= P_{Ext}cdot A_{top}$ and  $F_{bot}= P_{Ext}cdot A_{bot}$ the net force is:
$$F_{net}=F_{bot}- F_{top} =  P_{Ext}cdot (A_{bot}-A_{top})$$
Hence the difference.

Hydraulic lift connected to two pistons

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