After three half lives, the percentage of fraction of amount remain is a)35 b)12.5 c)50 d)75

Options

A. 0.01
B. 0.003
C. 0.03125
D. 0.002

Answer

To solve the problem of finding the percentage of the remaining amount after three half-lives, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life**:
- The half-life of a substance is the time required for half of the substance to decay or be transformed.

2. **Initial Amount**:
- Let the initial amount of the substance be \( n_0 \).

3. **Calculate the Amount After Each Half-Life**:
- **After 1 Half-Life**: The amount remaining is:
\[ \text{Amount after 1 half-life} = \frac{n_0}{2} \]
- **After 2 Half-Lives**: The amount remaining is:
\[ \text{Amount after 2 half-lives} = \frac{n_0}{2} \times \frac{1}{2} = \frac{n_0}{4} \]
- **After 3 Half-Lives**: The amount remaining is:
\[ \text{Amount after 3 half-lives} = \frac{n_0}{4} \times \frac{1}{2} = \frac{n_0}{8} \]

4. **Calculate the Fraction of the Amount Remaining**:
- The fraction of the initial amount that remains after three half-lives is:
\[ \text{Fraction remaining} = \frac{\text{Amount after 3 half-lives}}{\text{Initial amount}} = \frac{\frac{n_0}{8}}{n_0} = \frac{1}{8} \]

5. **Convert the Fraction to Percentage**:
- To find the percentage of the remaining amount, multiply the fraction by 100:
\[ \text{Percentage remaining} = \frac{1}{8} \times 100 = 12.5\% \]

### Final Answer:
The percentage of the amount remaining after three half-lives is **12.5%**.