What percentage of radioactive substance is left after 5 half lives? 

Options

A. `31%`
B. `3.125%`
C. `0.3%`
D. `1%`

Explanation

To determine the percentage of a radioactive substance left after 5 half-lives, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Half-Life**: The half-life of a radioactive substance is the time required for half of the substance to decay. After one half-life, 50% of the original substance remains.

2. **Formula for Remaining Substance**: The remaining amount of a radioactive substance after a certain number of half-lives can be expressed using the formula:
\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^n \]
where:
- \(N\) = remaining amount of substance
- \(N_0\) = initial amount of substance
- \(n\) = number of half-lives

3. **Substituting the Values**: In this case, we are interested in what happens after 5 half-lives, so we substitute \(n = 5\):
\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^5 \]

4. **Calculating the Fraction Remaining**: Now we calculate \(\left(\frac{1}{2}\right)^5\):
\[ \left(\frac{1}{2}\right)^5 = \frac{1}{32} \]

5. **Finding the Remaining Amount**: This means that after 5 half-lives, the remaining amount \(N\) is:
\[ N = N_0 \times \frac{1}{32} \]

6. **Calculating the Percentage Remaining**: To find the percentage of the original substance that remains, we multiply by 100:
\[ \text{Percentage remaining} = \left(\frac{N}{N_0}\right) \times 100 = \left(\frac{1}{32}\right) \times 100 \]

7. **Final Calculation**: Now, calculating \(\frac{100}{32}\):
\[ \frac{100}{32} = 3.125\% \]

### Conclusion:
After 5 half-lives, 3.125% of the radioactive substance remains.