I was given the following word problem:

If the half-life of a certain isotope is four years, what a fraction of a sample of that isotope will remain after 12 years?

This was simple for me. 12 years is equivalent to exactly 3 half-lives therefore 1/8 of the original sample amount remains.

But then I thought, what if a “non half-life” amount time passes? Not just like a different amount of years, but like months or days or something.

For example, if half of a half-life passes, does that mean 3/4 of the original sample remains?

• You are familiar with the exponential formula, right? – Cosmas Zachos Apr 19 at 21:10
• I forgot about this. Thanks for the refresher, it’s been a while – Ibby Apr 19 at 21:19

It is not a linear relationship, but a geometric (exponential) one, as you correctly suspected: after a time $$t$$ (expressed in the same unit as the half-life or period $$T$$), the remaining amount of non-decayed nuclei is:
$$N\ =\ N_0\ \frac{1}{2^{\left( t\ /\ T\right) }}$$