A radioactive nucleus can decay be either emitting an `alpha` particle or by emitting a `beta` particle. Probability of `alpha` decay is `75%` while that of `beta` decay is `25%` the decayconstatnt of `alpha` decay is `lamda_(1)` and that of `beta` decay is `lamda_(2)` is `(lamda_(1))/(lamda_(2))`

Explanation

To solve the problem, we need to find the ratio of the decay constants \(\lambda_1\) (for alpha decay) and \(\lambda_2\) (for beta decay) based on the probabilities of each type of decay.

### Step-by-Step Solution:

1. **Understand the Given Probabilities:**
- The probability of alpha decay, \(P(\alpha)\), is given as 75% or 0.75.
- The probability of beta decay, \(P(\beta)\), is given as 25% or 0.25.

2. **Relate Decay Constants to Probabilities:**
- The decay constant is related to the probability of decay. The ratio of the decay constants can be expressed in terms of the probabilities:
\[ \frac{\lambda_1}{\lambda_2} = \frac{P(\alpha)}{P(\beta)} \]

3. **Substitute the Values:**
- Substitute the values of the probabilities into the equation:
\[ \frac{\lambda_1}{\lambda_2} = \frac{0.75}{0.25} \]

4. **Calculate the Ratio:**
- Simplifying the fraction:
\[ \frac{0.75}{0.25} = 3 \]

5. **Final Result:**
- Therefore, the ratio of the decay constants is:
\[ \frac{\lambda_1}{\lambda_2} = 3 \]

### Conclusion:
The ratio of the decay constants \(\frac{\lambda_1}{\lambda_2}\) is 3.

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