How do you calculate the mechanical advantage of a compound pulley?

The mechanical advantage of a compound pulley equals the number of lines (e.g., ropes) that support all the movable pulleys in the system.
It almost seems magical that tremendous weights can be controlled by simply attaching a number of pulleys to them using a system or lines — like ropes or cables or wires — and pulling with a small amount for force. An arrangement like this is often called a “block and tackle” system.
The “block” refers to in the “block and tackle” refers to one or more pulleys, often called sheaves, mounted in a single assembly. Simple blocks may use one pulley, but for lifting very heavy weights many pulleys are often combined onto a single axle and mounted together with lines wound around them connecting the pulleys.
The whole assembly of several blocks and the lines that connect them is called the “tackle.”
To understand how a block and tackle (a.k.a., compound pulley) works, it’s only necessary to remember that the work required to move something is given by the force applied multiplied by the distance the object is moved.
If you just attach a line to an object and use a contstant force ##F## to move it a distance ##D##, the work is just ##F cdot D##.
If you loop the line around a single, fixed pulley (or sheave), all the pulley does is change the direction of motion so that the same, so that there is no real advantage and the work required to move the block a distance ##D## is the same: ##F cdot D##.
If, however, a moving sheave is attached to the object and the line is attached back to the fixed pulley, everything changes. Because the force is shared by the two lines attached to the moving sheave, the force exerted on the line held by the user is reduced by half. On the other hand, because the same total energy is required to move the object the full distance D, the lines must be pulled twice as far. This can be expressed mathmatically by noting that the total force ##F cdot D## (for a single fixed pulley, or no pulley) ##= F/2## (for the moving pulley with 2 lines attached attached) * ##2 D##.
This can be extended to multiple moving pulleys and lines: the force multiplier equals the number of lines attached to the moving pulley.
For more information on this, and other devices that provide mechanical advantage, check out the following link:
http://en.wikipedia.org/wiki/Mechanical_advantage