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

Options

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

Explanation

To solve the problem of finding the fraction of a sample of radium that would remain after 6400 years, given that the half-life period of radium is 1600 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Half-Life**: The half-life of a radioactive substance is the time required for half of the substance to decay. For radium, this is given as 1600 years.

2. **Determine the Number of Half-Lives**:
- We need to find out how many half-lives fit into the total time of 6400 years.
- Number of half-lives (n) = Total time / Half-life = 6400 years / 1600 years = 4.

3. **Use the Half-Life Formula**:
- The fraction of the remaining sample after n half-lives can be calculated using the formula:
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^n \]
- Substitute n = 4 into the formula:
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

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

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

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