Rock age by finding using elements and percentages

Question

Certain element has a half life of 2.5 billion years..15% of the original element remains. The other 85% decayed into lead. How old is the rock?

Analysis of another rock shows that it contains 65% of its original elements the other 35% decayed into lead. How old is ths rock?

Answer 1

fraction remaining = 0.5ⁿ where n = the number of half lives that have elapsed.

0.15 = 0.5ⁿ

log 0.15 = n log 0.5

-0.824 = -0.301n

n = 2.74 half lives have elapsed

2.74 x 2.5 billion years = 6.85 billion years = 6.9 billion years old

0.35 = 0.5ⁿ

log 0.35 = n log 0.5

-0.456 = 0.301 n

n = 0.51 half lives elapsed

0.51 x 2.5 billion years = 1.3 billion years old