Force Gain & Distance Loss: Explaining a Machine as Force Multiplier

[redXI]

"When a machine is used as force multiplier, the output force is larger than the input force. The force gain is paid by the distance loss."

I still don't get it. All I know is that this thing got to do with bikes? Correct me if I'm wrong.

 

[Galileo]

Reminds me of pulleys. You can lift a heavy object by pulling a rope hanging over some pulleys. You can lift the object with half the force needed without a pulley, but twice as much rope (length) will pass through your hands as the height gained by the object. This is simply because the total energy gained by the object (mgh) is equal to the work you have done W=Fd. You can make F smaller, but d becomes larger in the same proportion.
The same thing applies in a hydrolic lift, or a jack. You can lift heavy object by doing work on the lever, but you have to move the lever up and down maybe 20 times for the object (car of whatever) to raise half a meter.
In a bike the same sort of principle applies. If you shift in higher gear you have to push hard on the pedals, but you'll move a great distance in one cycle. If you're in a very low gear you have to cycle lots of times to get some push forwards, but it's much lighter.