What percentage (approximate) of original radioactive substance left after 4 half-Life?

Options

A. 0.08
B. 0.06
C. 0.04
D. 0.03

Answer

To determine the percentage of the original radioactive substance left after 4 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.

2. **Initial Amount**:
Let's denote the initial amount of the radioactive substance as \( A \).

3. **After 1 Half-Life**:
After one half-life, the amount remaining is:
\[ \text{Remaining after 1 half-life} = \frac{A}{2} \]

4. **After 2 Half-Lives**:
After two half-lives, the amount remaining is:
\[ \text{Remaining after 2 half-lives} = \frac{A}{2^2} = \frac{A}{4} \]

5. **After 3 Half-Lives**:
After three half-lives, the amount remaining is:
\[ \text{Remaining after 3 half-lives} = \frac{A}{2^3} = \frac{A}{8} \]

6. **After 4 Half-Lives**:
After four half-lives, the amount remaining is:
\[ \text{Remaining after 4 half-lives} = \frac{A}{2^4} = \frac{A}{16} \]

7. **Calculating the Percentage Left**:
To find the percentage of the original substance that remains, we use the formula:
\[ \text{Percentage remaining} = \left( \frac{\text{Remaining amount}}{\text{Original amount}} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage remaining} = \left( \frac{\frac{A}{16}}{A} \right) \times 100 = \frac{1}{16} \times 100 \]

8. **Final Calculation**:
\[ \text{Percentage remaining} = \frac{100}{16} = 6.25\% \]

### Conclusion:
After 4 half-lives, approximately **6.25%** of the original radioactive substance remains.