After a time equal to four half lives, the amount of radioactive material remaining undecayed is-

Options

A. `6.25%`
B. `12.5%`
C. `25%`
D. `50%`

Explanation

To solve the problem of finding the amount of radioactive material remaining undecayed after a time equal to four half-lives, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Half-Life Concept**: The half-life of a radioactive material is the time required for half of the radioactive atoms in a sample to decay. After one half-life, 50% of the original material remains.

2. **Determine the Amount Remaining After Each Half-Life**:
- After 1 half-life: \( n = \frac{n_0}{2} \)
- After 2 half-lives: \( n = \frac{n_0}{2^2} = \frac{n_0}{4} \)
- After 3 half-lives: \( n = \frac{n_0}{2^3} = \frac{n_0}{8} \)
- After 4 half-lives: \( n = \frac{n_0}{2^4} = \frac{n_0}{16} \)

3. **Calculate the Remaining Amount**: After 4 half-lives, the remaining amount of the radioactive material is:
\[ n = \frac{n_0}{16} \]

4. **Calculate the Percentage of Remaining Material**: To find the percentage of the original amount that remains undecayed, we use the formula:
\[ \text{Percentage remaining} = \left( \frac{n}{n_0} \right) \times 100 \]
Substituting the value of \( n \):
\[ \text{Percentage remaining} = \left( \frac{n_0/16}{n_0} \right) \times 100 = \frac{1}{16} \times 100 \]

5. **Perform the Calculation**:
\[ \frac{1}{16} \times 100 = 6.25\% \]

6. **Conclusion**: The amount of radioactive material remaining undecayed after four half-lives is **6.25%**.

### Final Answer:
The amount of radioactive material remaining undecayed after a time equal to four half-lives is **6.25%**.