When we look at sand in an hourglass, we can estimate how much time has passed based on the amount of sand that has fallen to the bottom.
Radioactive rocks offer a similar “clock.” Radioactive atoms, such as uranium (the parent isotopes), decay into stable atoms, such as lead (the daughter isotopes), at a measurable rate.
So what they can do is compare the amount that should be in whatever they're looking at into the amount that's left and using a formula which is the exponential decay formula, they can figure out how old something is.
Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.
To date a radioactive rock, geologists first measure the “sand grains” in the top glass bowl (the parent radioisotope, such as uranium-238 or potassium-40).
They also measure the sand grains in the bottom bowl (the daughter isotope, such as lead-206 or argon-40, respectively).
Each sample type has specific problems associated with its use for dating purposes, including contamination and special environmental effects.
More information on the sources of error in carbon dating are presented at the bottom of this page.
So, the scientist would find C14-to-C12 ratios ranging from: .34 \times 10^$ - to - [insert 000$ year calculation here].