Half lives of two radioactive nuclei `A` and `B` are `10` minutes and `20` minutes respectively, If initially a sample has equal number of nuclei, then after `60` minutes, the ratio of decayed numbers of nuclei `A` and `B` will be :

Options

A. `3 : 8`
B. `1 : 8`
C. `8 : 1`
D. `9 : 8`

Answer

To solve the problem, we need to determine the ratio of decayed nuclei of two radioactive substances A and B after 60 minutes, given their half-lives.

### Step-by-Step Solution:

1. **Identify the Half-Lives**:
- Half-life of nucleus A, \( t_{1/2}^A = 10 \) minutes
- Half-life of nucleus B, \( t_{1/2}^B = 20 \) minutes

2. **Calculate the Number of Half-Lives in 60 Minutes**:
- For nucleus A:
\[ \text{Number of half-lives for A} = \frac{60 \text{ minutes}}{10 \text{ minutes}} = 6 \]
- For nucleus B:
\[ \text{Number of half-lives for B} = \frac{60 \text{ minutes}}{20 \text{ minutes}} = 3 \]

3. **Determine the Remaining Nuclei After Each Half-Life**:
- Let the initial number of nuclei for both A and B be \( N_0 \).
- After 6 half-lives for A:
\[ N_A = N_0 \left( \frac{1}{2} \right)^6 = \frac{N_0}{64} \]
- After 3 half-lives for B:
\[ N_B = N_0 \left( \frac{1}{2} \right)^3 = \frac{N_0}{8} \]

4. **Calculate the Decayed Nuclei**:
- Decayed nuclei of A:
\[ \text{Decayed A} = N_0 - N_A = N_0 - \frac{N_0}{64} = N_0 \left(1 - \frac{1}{64}\right) = N_0 \left(\frac{63}{64}\right) \]
- Decayed nuclei of B:
\[ \text{Decayed B} = N_0 - N_B = N_0 - \frac{N_0}{8} = N_0 \left(1 - \frac{1}{8}\right) = N_0 \left(\frac{7}{8}\right) \]

5. **Calculate the Ratio of Decayed Nuclei**:
- The ratio of decayed nuclei A to decayed nuclei B:
\[ \text{Ratio} = \frac{\text{Decayed A}}{\text{Decayed B}} = \frac{N_0 \left(\frac{63}{64}\right)}{N_0 \left(\frac{7}{8}\right)} = \frac{\frac{63}{64}}{\frac{7}{8}} = \frac{63 \times 8}{64 \times 7} = \frac{504}{448} = \frac{63}{56} = \frac{9}{8} \]

### Final Answer:
The ratio of decayed numbers of nuclei A and B after 60 minutes is \( 9:8 \).