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

Options

A. `1 : 4`
B. `5 : 4 `
C. `1 : 16`
D. `4 : 1`

Explanation

To solve the problem, we need to determine the number of decayed nuclei of elements A and B after 80 minutes, given their half-lives.

### Step-by-Step Solution:

1. **Identify the Half-Lives**:
- The half-life of element A (T_A) = 20 minutes.
- The half-life of element B (T_B) = 40 minutes.

2. **Determine the Number of Half-Lives in 80 Minutes**:
- For element A:
\[ \text{Number of half-lives} = \frac{80 \text{ minutes}}{20 \text{ minutes}} = 4 \]
- For element B:
\[ \text{Number of half-lives} = \frac{80 \text{ minutes}}{40 \text{ minutes}} = 2 \]

3. **Calculate the Remaining Nuclei after 80 Minutes**:
- Let the initial number of nuclei for both A and B be \( N_0 \).

- For element A:
\[ N_A = N_0 \left(\frac{1}{2}\right)^{4} = N_0 \times \frac{1}{16} \]
- Thus, the number of decayed nuclei of A:
\[ \text{Decayed A} = N_0 - N_A = N_0 - \frac{N_0}{16} = \frac{15N_0}{16} \]

- For element B:
\[ N_B = N_0 \left(\frac{1}{2}\right)^{2} = N_0 \times \frac{1}{4} \]
- Thus, the number of decayed nuclei of B:
\[ \text{Decayed B} = N_0 - N_B = N_0 - \frac{N_0}{4} = \frac{3N_0}{4} \]

4. **Calculate the Ratio of Decayed Nuclei of A to B**:
- The ratio of decayed nuclei of A to B is:
\[ \text{Ratio} = \frac{\text{Decayed A}}{\text{Decayed B}} = \frac{\frac{15N_0}{16}}{\frac{3N_0}{4}} \]
- Simplifying this:
\[ = \frac{15N_0}{16} \times \frac{4}{3N_0} = \frac{15 \times 4}{16 \times 3} = \frac{60}{48} = \frac{5}{4} \]

### Final Answer:
The ratio of decayed numbers of A to B nuclei after 80 minutes is \( \frac{5}{4} \).

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