The radioactive isotope carbon-14 is found in living material in small quantities. There are
approximately 1000 carbon-14 atoms for every 10 15 carbon-12 atoms. Whilst the material is still
living this ratio remains constant, because even though the carbon-14 is decaying, it is being
constantly replenished. When the material dies the carbon-14 decays and is not replaced. The
half-life of carbon-14 is about 6000 years.
In a bone the ratio of carbon-14 to carbon-12 atoms is found to be 100 : 10 15 .
Which of the following is the closest estimate of the age of the bone?

Answer:

27) E

When the material is living the ratio is 1000 : 10¹⁵

When the material died, the ration became 100 : 10¹⁵

This means that the ratio decreased by a factor of 10.

1 half-life = 6000 years

Let number of half-lives be M

¹⁰⁰⁰/₁₀₀ = 2^M

10 = 2^M

We know 2³ = 8 and 2⁴ = 16

So number of half-lives is a decimal between 3 and 4. (Let us take it as 3.5)

Number of half-lives = 3.5
Number of years = 6000 x 3.5 = 21,000 years
This is close to 20,000 so E is the answer.