The half life of `Ra` is 1600 years. The fraction of a sample of `Ra` that would remain after 6400 years is

Options

A. `1//16`
B. `1//4`
C. `1//8`
D. `1//2`

Explanation

To solve the problem of determining the fraction of a sample of radium (Ra) that remains after 6400 years, given its half-life of 1600 years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the half-life**: The half-life of radium (Ra) is given as 1600 years.

2. **Determine the total time elapsed**: We need to find out how much time has passed, which is given as 6400 years.

3. **Calculate the number of half-lives**: To find out how many half-lives fit into the total time, we divide the total time by the half-life:
\[ \text{Number of half-lives} (n) = \frac{\text{Total time}}{\text{Half-life}} = \frac{6400 \text{ years}}{1600 \text{ years}} = 4 \]

4. **Use the formula for remaining fraction**: The fraction of the sample that remains after \( n \) half-lives can be calculated using the formula:
\[ \text{Fraction remaining} = \left(\frac{1}{2}\right)^n \]
Substituting \( n = 4 \):
\[ \text{Fraction remaining} = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

5. **Conclusion**: Therefore, the fraction of the sample of radium that would remain after 6400 years is \( \frac{1}{16} \).

### Final Answer:
The fraction of a sample of Ra that would remain after 6400 years is \( \frac{1}{16} \).

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