The half life of a radio isotope is 5 years . The fraction which will decay in 15 years , will be

Options

A. `1//16`
B. `3//4`
C. `7//8`
D. `5//8`

Explanation

To solve the problem, we need to determine the fraction of a radioactive isotope that decays over a period of 15 years, given that its half-life is 5 years.

### Step-by-Step Solution:

1. **Identify the Half-Life**:
The half-life (T_half) of the radioactive isotope is given as 5 years.

2. **Determine the Number of Half-Lives in 15 Years**:
To find out how many half-lives fit into 15 years, we divide the total time by the half-life:
\[ \text{Number of half-lives} = \frac{15 \text{ years}}{5 \text{ years}} = 3 \]
This means that in 15 years, 3 half-lives have occurred.

3. **Calculate Remaining Quantity After Each Half-Life**:
- After the first half-life (5 years), the remaining quantity is:
\[ \frac{x}{2} \]
- After the second half-life (10 years), the remaining quantity is:
\[ \frac{x}{2} \times \frac{1}{2} = \frac{x}{4} \]
- After the third half-life (15 years), the remaining quantity is:
\[ \frac{x}{4} \times \frac{1}{2} = \frac{x}{8} \]

4. **Calculate the Decayed Quantity**:
The initial quantity of the radioactive isotope is \( x \). After 15 years, the remaining quantity is \( \frac{x}{8} \). Therefore, the quantity that has decayed is:
\[ \text{Decayed quantity} = x - \frac{x}{8} = \frac{8x}{8} - \frac{x}{8} = \frac{7x}{8} \]

5. **Determine the Fraction Decayed**:
The fraction that has decayed is given by the decayed quantity divided by the initial quantity:
\[ \text{Fraction decayed} = \frac{\frac{7x}{8}}{x} = \frac{7}{8} \]

### Final Answer:
The fraction of the radioactive isotope that will decay in 15 years is \( \frac{7}{8} \).

---