if the big piston is 10 times of the sizes of the smaller piston to balance a 500Nload at the big piston (assume that the piston level are the same and the weight of the big piston can be ignored

consider two pistons, A and B

ForceA/areaA=forceB/areaB
in your case, AreaA/AreaB = 10 as I read it.

To balance a load of 500N at the big piston, we need to determine the force required at the smaller piston. Let's set up the equation:

Force at big piston = Force at small piston

Given that the big piston is 10 times the size of the smaller piston, we can express their sizes as a ratio:

Size of big piston : Size of small piston = 10 : 1

However, since the size affects the force exerted, the ratio of their sizes also applies to the force:

Force at big piston : Force at small piston = 10 : 1

Now we can write the equation using this ratio:

500N : Force at small piston = 10 : 1

To find the force at the small piston, we need to solve for it. We can do this by cross-multiplying and simplifying the equation:

500N * 1 = Force at small piston * 10

Simplifying further:

500N = 10 * Force at small piston

Dividing both sides by 10:

Force at small piston = 500N / 10

The force at the small piston is therefore:

Force at small piston = 50N

Hence, a force of 50N at the smaller piston will balance a load of 500N at the big piston.