Half - lives of two radioactive elements A and B are 20 minutes and 40 minutes respectively. Initially, The samples have equal number of nuclie. After `80` minutes ,the ratio of decayed number of `A and B` nuclei will be

Explanation

To find the ratio of the decayed number of nuclei of radioactive elements A and B after 80 minutes, we can follow these steps:

### Step 1: Determine the decay constants for A and B
The decay constant \( \lambda \) is given by the formula:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
For element A, with a half-life \( t_{1/2} = 20 \) minutes:
\[ \lambda_A = \frac{\ln(2)}{20 \text{ min}} \]
For element B, with a half-life \( t_{1/2} = 40 \) minutes:
\[ \lambda_B = \frac{\ln(2)}{40 \text{ min}} \]

### Step 2: Calculate the remaining nuclei of A after 80 minutes
Using the formula for the remaining nuclei:
\[ N = N_0 e^{-\lambda t} \]
For element A after \( t = 80 \) minutes:
\[ N_A = N_0 e^{-\lambda_A \cdot 80} \]
Substituting \( \lambda_A \):
\[ N_A = N_0 e^{-\left(\frac{\ln(2)}{20}\right) \cdot 80} \]
This simplifies to:
\[ N_A = N_0 e^{-4 \ln(2)} = N_0 \cdot \left(2^{-4}\right) = \frac{N_0}{16} \]

### Step 3: Calculate the decayed number of nuclei of A
The decayed number of nuclei of A is:
\[ N_A' = N_0 - N_A = N_0 - \frac{N_0}{16} = \frac{15N_0}{16} \]

### Step 4: Calculate the remaining nuclei of B after 80 minutes
For element B after \( t = 80 \) minutes:
\[ N_B = N_0 e^{-\lambda_B \cdot 80} \]
Substituting \( \lambda_B \):
\[ N_B = N_0 e^{-\left(\frac{\ln(2)}{40}\right) \cdot 80} \]
This simplifies to:
\[ N_B = N_0 e^{-2 \ln(2)} = N_0 \cdot \left(2^{-2}\right) = \frac{N_0}{4} \]

### Step 5: Calculate the decayed number of nuclei of B
The decayed number of nuclei of B is:
\[ N_B' = N_0 - N_B = N_0 - \frac{N_0}{4} = \frac{3N_0}{4} \]

### Step 6: Calculate the ratio of decayed nuclei of A to B
Now, we can find the ratio of the decayed nuclei:
\[ \text{Ratio} = \frac{N_A'}{N_B'} = \frac{\frac{15N_0}{16}}{\frac{3N_0}{4}} = \frac{15}{16} \cdot \frac{4}{3} = \frac{15 \cdot 4}{16 \cdot 3} = \frac{60}{48} = \frac{5}{4} \]

### Final Answer
The ratio of the decayed number of A and B nuclei after 80 minutes is:
\[ \frac{5}{4} \]