The half-life of a radioactive isotope `X` is `20 yr`. It decays to another element `Y` which is stable. The two elements `X` and `Y` were found to be in the ratio `1:7` in a sample of given rock. The age of the rock is estimated to be

Options

A. 40 yr
B. 60 yr
C. 80 yr
D. 100 yr

Explanation

To solve the problem, we need to determine the age of the rock based on the given half-life of the radioactive isotope X and the ratio of X to Y found in the rock sample.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The half-life of isotope X is \( t_{1/2} = 20 \) years.
- The ratio of elements X to Y found in the rock is \( 1:7 \).

2. **Set Up the Mass Relationship:**
- Let the mass of X be \( m_X \) and the mass of Y be \( m_Y \).
- According to the ratio, we can express the masses as:
\[ m_Y = 7 m_X \]

3. **Total Mass Calculation:**
- The total mass of the sample can be expressed as:
\[ m = m_X + m_Y \]
- Substituting \( m_Y \) into the equation gives:
\[ m = m_X + 7 m_X = 8 m_X \]
- Therefore, we can express \( m_X \) in terms of total mass \( m \):
\[ m_X = \frac{m}{8} \]
- And for \( m_Y \):
\[ m_Y = 7 \cdot \frac{m}{8} = \frac{7m}{8} \]

4. **Determine the Remaining Fraction of X:**
- The total amount of X that remains is \( m_X \), and the amount of Y formed is \( m_Y \).
- Initially, all of the mass was X, so after some time, the remaining fraction of X is:
\[ \text{Remaining fraction of } X = \frac{m_X}{m} = \frac{\frac{m}{8}}{m} = \frac{1}{8} \]

5. **Calculate the Number of Half-Lives:**
- The remaining fraction of X is \( \frac{1}{8} \). The decay process can be summarized as follows:
- After 1 half-life: \( \frac{1}{2} \)
- After 2 half-lives: \( \frac{1}{4} \)
- After 3 half-lives: \( \frac{1}{8} \)
- This shows that it takes 3 half-lives to reduce the amount of X to \( \frac{1}{8} \) of its original amount.

6. **Calculate the Age of the Rock:**
- The age of the rock can be calculated as:
\[ \text{Age} = n \cdot t_{1/2} \]
- Where \( n \) is the number of half-lives (which is 3), and \( t_{1/2} = 20 \) years:
\[ \text{Age} = 3 \cdot 20 \text{ years} = 60 \text{ years} \]

### Final Answer:
The age of the rock is **60 years**.