Question: What is the advantage for rockets to [have multiple stages](http://en.wikipedia.org/wiki/Multistage_rocket)? 

Wouldn't a single stage with the same amount of fuel weigh less?

**Note** I would like a quantitative answer, if possible :-)

![alt text](https://i.stack.imgur.com/lswdV.jpg)

Answer: 
Launch weight would be lower if you had a fixed fuel load, and only one engine and fuel tank. However the specific impulse applied to the payload would be lower. The problem is that even when the fuel is say 90% exhausted, the rocket is still trying to accelerate the now grossly oversized fuel tank and engine. So the trick is to try to reduce the deadweight (structural mass) as the fuel is consumed. Another compromise system is to have jetisonable external fuel tanks, like the shuttle, which are thrown away once their fuel is consumed.
Another aspect to consider is the burn characteristics of the rocket motors. This is especially important in solid rocket motors because once lit, they are self-oxidizing and are not easy to turn off. 

At low altitudes, the rocket should not accelerate too rapidly because the air is very dense and the power required is proportional to velocity cubed. So you just want to get the thing going until you reach an altitude where the air is less dense and it's more economical to go fast. So you may have a first stage that burns relatively slowly. 

Once that burns and you reach a higher altitude where you can go faster, you drop your "slow burn" motor and kick on your powerful motor. Now the air density is much lower so you can accelerate as quickly as you want and reach whatever speed is needed. 

A third stage might be used to fine-tune the speed and position the craft in whatever orbit or trajectory is needed. 

Furthermore, because the air pressure decreases with altitude, the ideal nozzle shape is not the same at low altitude as high. An [overexpanded nozzle](http://en.wikipedia.org/wiki/Rocket_engine_nozzle#Aerostatic_back-pressure_and_optimum_expansion) at sea-level can be an underexpanded nozzle at altitude resulting in a very narrow range where it is operating at maximum efficiency. 

You could design adaptive nozzles but they are very heavy and expensive, and they can't really be made to service the full range. Or, you could have stages with fixed nozzles that are designed to be as efficient as possible over the range of altitudes serviced. 

So, in addition to the answers above about dropping weight as you go resulting in less fuel use, each stage can also be designed to take into account the operating regime and needs by carefully selecting the fuel for appropriate thrust/burn rates and by designing the nozzle for the nominal operating conditions for the altitudes serviced by the motor. Both of which lead to a much more efficient motor and thus less fuel.