After 3 half-lives, what percentage of the original number of parent atoms is left?  _______ (a number)

1. ### Break Down the Problem

To determine the percentage of the original number of parent atoms remaining after 3 half-lives, we need to understand how the concept of half-lives works.

2. ### Relevant Concepts

The concept of half-life states that after each half-life, half of the remaining parent atoms decay into daughter atoms. The formula to calculate the remaining quantity after $n$ half-lives is:

$N = N_0 \left(\frac{1}{2}\right)^n$

where $N_0$ is the original number of parent atoms, $n$ is the number of half-lives, and $N$ is the remaining number of parent atoms.

3. ### Analysis and Detail

1. For $n = 3$ (3 half-lives), the remaining proportion of parent atoms is calculated as: $N = N_0 \left(\frac{1}{2}\right)^3 = N_0 \left(\frac{1}{8}\right)$

2. To find the percentage of the original atoms left, we convert the fraction to a percentage: $\text{Percentage Remaining} = \left(\frac{N}{N_0}\right) \times 100 = \left(\frac{1}{8}\right) \times 100 = 12.5\%$

4. ### Verify and Summarize

We calculated the percentage of the original parent atoms remaining after 3 half-lives, confirming that after 3 half-lives, only $12.5\%$ of the original number of parent atoms is left.

Final Answer

12.5