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

Options

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

Explanation

To solve the problem, we will use the concept of radioactive decay and the relationship between the amounts of the radioactive isotope and its decay product.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- The half-life of isotope \( X \) is \( 20 \) years.
- The ratio of the amounts of \( X \) and \( Y \) is \( 1:7 \).
- This means for every \( 1 \) part of \( X \), there are \( 7 \) parts of \( Y \).

2. **Setting Up the Initial Conditions**:
- Let the initial amount of \( X \) be \( N_0 \).
- After some time \( t \), the amount of \( X \) remaining will be \( N_X \) and the amount of \( Y \) formed will be \( N_Y \).
- From the ratio given, we have:
\[ \frac{N_X}{N_Y} = \frac{1}{7} \]
- This implies:
\[ N_Y = 7N_X \]

3. **Total Initial Amount**:
- The total initial amount of the sample (before decay) is:
\[ N_0 = N_X + N_Y = N_X + 7N_X = 8N_X \]

4. **Using the Decay Formula**:
- The decay of the radioactive isotope can be described by the equation:
\[ N_X = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
- Where \( T_{1/2} = 20 \) years is the half-life of \( X \).

5. **Substituting into the Ratio**:
- We can express \( N_Y \) in terms of \( N_0 \):
\[ N_Y = N_0 - N_X = 8N_X - N_X = 7N_X \]
- Thus, we can write:
\[ \frac{N_X}{N_0} = \frac{N_X}{8N_X} = \frac{1}{8} \]
- Therefore, we can equate this to the decay formula:
\[ \frac{1}{8} = \left( \frac{1}{2} \right)^{\frac{t}{20}} \]

6. **Solving for Time \( t \)**:
- Since \( \frac{1}{8} = \left( \frac{1}{2} \right)^3 \), we can write:
\[ \left( \frac{1}{2} \right)^3 = \left( \frac{1}{2} \right)^{\frac{t}{20}} \]
- By comparing the exponents, we have:
\[ 3 = \frac{t}{20} \]
- Thus, solving for \( t \):
\[ t = 3 \times 20 = 60 \text{ years} \]

### Conclusion:
The age of the rock is estimated to be **60 years**.