Half life period of a radioactive substance is 1600 minutes . What fraction of the substance will remain after 6400 minutes :

Options

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

Explanation

To solve the problem of how much of a radioactive substance remains after a certain period, we can use the concept of half-lives. Here’s a step-by-step solution:

### Step 1: Understand the concept of half-life
The half-life of a radioactive substance is the time required for half of the substance to decay. In this case, the half-life is given as 1600 minutes.

### Step 2: Determine the total time elapsed
We need to find out how much of the substance remains after 6400 minutes.

### Step 3: Calculate the number of half-lives that have passed
To find out how many half-lives fit into 6400 minutes, we divide the total time by the half-life:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{6400 \text{ minutes}}{1600 \text{ minutes}} = 4 \]

### Step 4: Calculate the remaining fraction of the substance
After each half-life, the remaining amount of substance is halved. Therefore, after 4 half-lives, the remaining fraction can be calculated as:
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^n \]
where \( n \) is the number of half-lives. In this case, \( n = 4 \):
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

### Conclusion
After 6400 minutes, the fraction of the radioactive substance that will remain is \( \frac{1}{16} \).

---