As I learn more about the design of multi stage rockets I try to pass the new information on to you. Recently I've been reading more about the Minimum Cost Design system of Arthur Schnitt. Although the actual report on his method is still restricted, probably to keep the public from learning how cheap rockets can really be, I have picked out a few relevant points.
In a multi stage rocket the stages get smaller as you go up. The ratio between stages is usually between 3 to 5. Depending on the mission requirements you can have high acceleration or high efficiency in determining the size of the booster. For an ICBM which needs to leave the launch site in a big hurry, high acceleration is best, you use big engines and small tanks. For high efficiency, you want to run the booster engines as long as possible to keep velocity low in the dense lower atmosphere, use bigger tanks. The staging ratio is also the reason that you invest more technology in the upper stages. The Saturn 5 had a stage ratio of 4.5 for it's first and second stages and I'll use it as an example. Since the second stage is the payload of the first stage, adding a pound to the second stage means that you must add 4.5 pounds more to the booster to lift that additional pound of second stage, this is called the growth factor. Since every pound removed from the second stage saves 4.5 ponds of first stage, it makes sense to use higher technology on the upper stages or conversely lower technology in the lower stages. The payload should share the same technology as the last stage. Because the upper stages do not have to provide the initial liftoff momentum and can actually loose altitude while building up horizontal velocity, the upper stages can have proportionally less thrust than the first stage. Typical ratios are 6 to 8 for first and second stage thrust. Of course, once in orbit thrust becomes irrelevant and ISP rules the day.
At the salvage yard where I hunt for materials to build rockets, the cost of steel is about 12 cents per pound and aluminum is about 70 cents per pound, a ratio of about 5.83, pretty close to the stage ratio, which is why our second stage is now made of aluminum.
I never understood why the Apollo program went with Lunar Orbit Rendezvous, well as I recently found out, the reason is, by leaving in lunar orbit the Apollo capsule reentry heat shield and the fuel necessary to send the Apollo capsule back to Earth from Lunar orbit, and not landing those two items on the Moon and lifting them off the Moon and back into Lunar orbit, enough propellant mass was saved to more than pay for the use of a separate Lunar Lander and to throw it away. The original plan of Direct Ascent called for the Apollo capsule to land on the Moon, and the capsule and service module were designed with that in mind. If Lunar Orbit Rendezvous had entered the picture at the start of the Apollo program the design of the capsule and lander might have been slightly different, possibly more like the Soviet Soyuz system.