Based on its carbon-14 to carbon-12 ratio, a fossil is 11,500 years old. Using the half-life of carbon-14 (5730 years), what percentage of the original carbon-14 remains?

What you are being asked

You are given an age and a half-life, and you need the fraction of the original carbon-14 that is still present. This is a standard exponential decay (half-life) calculation.

Using the half-life decay model

For a substance with half-life $T_{1/2}$, the amount remaining after time $t$ is $$\text{fraction remaining}=\left(\frac{1}{2}\right)^{t/T_{1/2}}$$ Here, $t=11{,}500$ years and $T_{1/2}=5730$ years: $$\text{fraction remaining}=\left(\frac{1}{2}\right)^{11500/5730}$$

Computing the exponent and percentage

First find how many half-lives have passed: $$\frac{t}{T_{1/2}}=\frac{11500}{5730}\approx 2.00698$$ Now evaluate: $$\left(\frac{1}{2}\right)^{2.00698}\approx 0.249$$ Convert to a percent: $$0.249\times 100\%\approx 24.9\%$$ So it is just under two half-lives, which is why the result is slightly less than $25\%$.

Quick reasonableness check

After 1 half-life, $50\%$ remains; after 2 half-lives, $25\%$ remains. Since $11{,}500$ years is a tiny bit more than $2\times 5730=11{,}460$ years, the remaining percent should be a bit less than $25\%$, matching $24.9\%$.