The naturally occurring radioisotopes uranium-238 and uranium-235 have decay chains that end with the stable isotopes lead-206 and lead-207, respectively. Natural minerals such as zircons contain these uranium and lead isotopes. Careful measurements of the relative amounts of the isotopes can be used to estimate the ages of such minerals.

(a) How many separate alpha decays are there in each decay chain?

(b) For a mineral sample that is 4.5 billion years old, what would be the approximate ratio of the number of uranium-238 nuclei to the number of lead-206 nuclei? (Assume the initial decay of uranium-238 is the slowest in the decay chain.) 

(c) Repeat part (b) for uranium-235 and lead-207.
(d) If you could only measure the ratio of the number of lead-206 nuclei to the number of lead-207 nuclei, what would the approximate value be, assuming the normal relative abundance of uranium-238 and uranium-235?