Two radioactive substances X and Y originally have `N_(1) and N_(2)` nuclei respectively. Half life of X is half of the half life of Y. After three half lives of Y, number of nuclei of both are equal. `(N_(1))/(N_(2))` The ratio N will be equal to :

Options

A. `(1)/(8)`
B. `(3)/(1)`
C. `(8)/(1)`
D. `(1)/(3)`

Answer

To solve the problem, we need to analyze the decay of two radioactive substances, X and Y, given their half-lives and initial quantities.

### Step-by-Step Solution:

1. **Understanding Half-lives**:
- Let the half-life of substance Y be \( T \).
- Then, the half-life of substance X is \( \frac{T}{2} \).

2. **Initial Nuclei**:
- Let the initial number of nuclei of substance X be \( N_1 \).
- Let the initial number of nuclei of substance Y be \( N_2 \).

3. **Decay After Three Half-lives of Y**:
- After 3 half-lives of Y (which is \( 3T \)):
- The number of nuclei of Y remaining will be:
\[ N_Y = \frac{N_2}{2^3} = \frac{N_2}{8} \]
- The time elapsed for substance X during this period is \( 3T \), which corresponds to \( 6 \) half-lives of X (since \( \frac{3T}{\frac{T}{2}} = 6 \)):
- The number of nuclei of X remaining will be:
\[ N_X = \frac{N_1}{2^6} = \frac{N_1}{64} \]

4. **Setting the Nuclei Equal**:
- According to the problem, after 3 half-lives of Y, the number of nuclei of both substances is equal:
\[ N_X = N_Y \]
- Substituting the expressions we derived:
\[ \frac{N_1}{64} = \frac{N_2}{8} \]

5. **Cross-Multiplying to Find the Ratio**:
- Cross-multiplying gives:
\[ 8N_1 = 64N_2 \]
- Dividing both sides by \( N_2 \):
\[ \frac{N_1}{N_2} = \frac{64}{8} = 8 \]

6. **Final Result**:
- Therefore, the ratio \( \frac{N_1}{N_2} \) is:
\[ \frac{N_1}{N_2} = 8 \]

### Conclusion:
The ratio \( \frac{N_1}{N_2} \) is equal to **8**.

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