These are two very simple questions to answer; one depends on chemistry, the other on "basic" mathematics:

not all lead come from uranium, as they produce naturally throughout the environment. I had someone told me this would make lead unreliable. However, in this case, the half life of uranium will be used since there is no new sources to replenish uranium in the sample. Thus, the remaining of uranium will help determine the age of the sample |

Whether or not lead can be formed by other sources is rather irrelevant.

Uranium-lead dating works by comparing the current ratio of uranium to lead, because of the simple fact that for every uranium atom decaying, you form one lead atom. At the beginning of the process there were no lead atoms in the structure (because we don't just scoop up a handful of dirt and stick it in a geigercounter). The measurements are (usually) made on zircons, a zirconium-silicate mineral with a formula ZrSiO

_{4}. When this mineral forms from molten rock, there is the chance of uranium sneaking into the structure of the mineral instead of zirconium, which gets trapped there after the mineral cools. But lead cannot enter the structure anywhere near as easily; it just isn't the right size. So virtually no lead is initally trapped in the mineral and we can safely assume that the initial amount is zero lead.

So, any lead present must have been formed by decay. Thus if there is a 1:1 ratio, half of the uranium has decayed, meaning one halflife has passed since the formation of the rock; if the ration is 1:3 then three-quarters of the uranium has decayed and there has passed two halflives -- the progression is in powers of 0.5 or 1/2; after *n* halflives there is 100 x 0.5^{n}% of the original isotope remaining.

All we have to do is compare the ratio in the manner above to determine the number of halflives since the mineral's formation. It does get a little more complicated because the fact that not one, but two uranium isotopes might be involved: uranium-235 and uranium-238. There's a method that involves something called concordia, but I'll leave that to more knowledgeable persons.

"But wait!" you say. "How do we know what the halflife is?!" Excellent question and one that has been answered hundreds of times here already. But just for kicks, and because I want to try explaining it myself (the best way to tets your knowledge is teach someone else):

It takes a very long time for uranium to decay into lead. Who even lived that long to observe this? If this is not how they determine the half life, then in what method do they use to determine the half life? I think it has to do with calculations. |

You are correct in surmising that it involve calculations.

Isotope decay follows an exponential decay pattern; the amount of material remaining at time t can be calculated by

**A**_{t} = A_{0}e^{kt} where A

_{0} is the original amount (amount at time zero), e is the natural base (about 2.7-something-or-other IIRC), k is a constant specific to that isotope (and can be worked out from an experiment by doing some fancy jiggery-pokery with calculus) and t is simply enough time (in whatever units; in this case it would be years although some do work in days, minutes, even milliseconds. Gotta be careful with that).

By doing an experiment where we begin with say 100 grams* of uranium and watching it decay over a period of 1 year, for example, we measure that there is now 99.7 grams* of uranium left. That tells us that A_{1} = 99.7.

So using our equation, **99.7 = 100 x e**^{k x 1}. Using the natural logarithm we can juggle the stuff until we get k on its lonesome:

99.7 = 100e^{k}

99.7/100 = e^{k}

0.997 = e^{k}

*ln*(0.997) = *ln* (e^{k}) = k = ???

All you need to do is find out what *ln*(0.997 is) to get k, which you can then place into the formula, and use that to work out the amount at any time you like OR figure out how much was in the original amount when given the amount at a later point.

This formula and experiment works on any isotope that can be measured accurately, not just uranium, and the general method can be extended into multiple areas of science like biology (simple population growth modelling) and physics (Newton's law of cooling).

You can find out most of this by looking up the subject in Google or checking out pretty much anything RAZD has written in the "Dates And Dating" forum.

* These are just random numbers. 1s are easy to manipulate.

** *ln* is notation for the natural log or logarithm to base e (normal logs use base 10).