The half lives of radioactive elements `X` and `Y` are `3` mintue and `27` minute respectively. If the activities of both are same, then calculate the ratio of number of atoms of X and Y.

Explanation

To solve the problem, we need to find the ratio of the number of atoms of radioactive elements X and Y, given that their activities are the same and their half-lives are different.

### Step-by-Step Solution:

1. **Understand the Concept of Activity**:
The activity (A) of a radioactive substance is defined as the number of disintegrations per unit time. It is directly proportional to the number of active nuclei (n) and the decay constant (λ):
\[ A = \lambda n \]

2. **Set Up the Equation for Activities**:
Since the activities of elements X and Y are the same, we can write:
\[ A_X = A_Y \]
This implies:
\[ \lambda_X n_X = \lambda_Y n_Y \]

3. **Relate Decay Constant to Half-Life**:
The decay constant (λ) is related to the half-life (t_half) by the formula:
\[ \lambda = \frac{0.693}{t_{half}} \]
Thus, we can express λ for both elements:
\[ \lambda_X = \frac{0.693}{t_{half, X}} \quad \text{and} \quad \lambda_Y = \frac{0.693}{t_{half, Y}} \]

4. **Substitute the Half-Lives**:
Given that the half-life of X is 3 minutes and that of Y is 27 minutes, we can substitute these values:
\[ \lambda_X = \frac{0.693}{3} \quad \text{and} \quad \lambda_Y = \frac{0.693}{27} \]

5. **Substitute λ into the Activity Equation**:
Now, substituting λ into the activity equation gives:
\[ \frac{0.693}{3} n_X = \frac{0.693}{27} n_Y \]

6. **Cancel Out Common Terms**:
The 0.693 cancels out from both sides:
\[ \frac{n_X}{3} = \frac{n_Y}{27} \]

7. **Rearranging the Equation**:
Rearranging the equation to find the ratio of the number of atoms:
\[ \frac{n_X}{n_Y} = \frac{3}{27} \]

8. **Simplify the Ratio**:
Simplifying the ratio gives:
\[ \frac{n_X}{n_Y} = \frac{1}{9} \]

### Final Answer:
The ratio of the number of atoms of X to Y is:
\[ \frac{n_X}{n_Y} = \frac{1}{9} \]