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.
The easiest way to think of it is this, imagine all the mass left over when a rocket has burned 85% of it's fuel. The mass of most of the tank and structure is now overkill and waste. It would be nice to be able to jettison that extra mass so that the fuel left can accelerate only the payload.

That's what a multi-stage rocket does. It jettisons the mass of initial stages so that the remaining fuel and thrust can accelerate much smaller mass to a much higher velocity than it would have been able to if there was only one stage. Remember acceleration is proportional to mass, so if you can get rid of say 80% of the mass then you can accelerate the payload 5 times more for the same remaining fuel.

Another benefit is that you can use rocket motors that are tuned for different velocities. In the initial stage you need maximum thrust and the rocket is not moving as fast. In the later stages you want high efficiency motors, not necessarily high thrust.

To get very high velocities it requires less overall fuel and mass with multiple stages. This comes at the cost of greater complexity and cost.