# Question #ab040

##### 1 Answer

The answer is **(B)**

#### Explanation:

A radioactive isotope's *half-life* tells you the time needed for an initial sample of said isotope to be **halved**.

More specifically, an initial sample of a radioactive element will be **halved** for *every* half-life that passes. If you start at

#100% -> t = 0#

#50% -> t = t_"1/2"#

#25% -> t = 2 * t_"1/2"#

#12.5% -> t = 3 * t_"1/2"#

#6.25% -> t = 4 * t_"1/2"#

This is equivalent to saying that

#"what you have" = "what you started with"/2^n" "# , where

*the number of half-lives that passed*.

In your case, you start with a sample of **100 mg**. In order for the sample to be reduced to **25 mg**, you need to have

#"100 mg" -> t = 0#

#"50 mg" -> t = t_"1/2"#

#"25 mg" -> t = 2 * t_"1/2"#

Your remaining sample is now *a quarter* the size it was in the beginning, which can only mean that *two half-lives* have passed

#"25 mg" = "100 mg"/2^n#

#2^n = (100color(red)(cancel(color(black)("mg"))))/(25color(red)(cancel(color(black)("mg")))) = 4 implies n = color(green)(2)#

The time that passed is thus equal to

#t = 2 * t_"1/2" = 2 * "5760 years" = color(green)("11520 years")#