By some simple algebraic manipulation of the basic radioactivity formula above, one can show that the following formula must hold at any time t: (Sr87/Sr86) is the ratio of these two isotopes at time t.
Note that this equation is in the simple form y = b m x, namely the formula for graph of a straight line with slope m and with y-intercept b: here y = (Sr87/Sr86).
Fortunately, scientists have developed several methods that not only circumvent the difficulty of not knowing the original amounts, but also provide a very reliable means of statistical validity checking.
For example, the rubidium-strontium isochron method, one of the most widely used schemes, is based on the radioactive decay of rubidium-87 into strontium-87 by the emission of a high-energy electron.
If all we have is one data point, the formula above doesn't help much more than the original formula.
But if we have multiple data points -- multiple measurements of different samples say within a single igneous rock, then these should all lie on a straight line, whose slope m is simply related to the age of the specimen by the formula m = e; instead, this original ratio actually comes out as a result of the calculation!
Also, such a calculation does not provide us with any statistical error margin to double-check the result.
Radiometric dating is rooted in the rates of radioactive decay of various isotopes, which rates have been measured carefully in numerous laboratories beginning in the early 20th century.
Radioactive decay is in turn a very basic physical phenomenon, well understood as a consequence of quantum mechanics.
Quantum mechanics is one of two cornerstones of modern physics (the other is general relativity), and has been precisely confirmed in thousands of very exacting experiments.
For these reasons, scientists have considerable confidence in these dates when they are measured properly in accordance with procedures that have been developed and refined over several decades.