Three isotopes fo an elecment have mass numbers ` (m, (m + 1)` and `(M + 2)` . If the mean mass number is ( M + 0. 5) then which of the following ratios may be accepted for ` M`, `(M + 1)` and `( M+2)` in tha order ?

Options

A. `1: 1:1
B. `4:1:1
C. `3:2:1`
D. `2:1:1

Explanation

To solve the problem, we need to find the acceptable ratios of the isotopes with mass numbers \( M \), \( M + 1 \), and \( M + 2 \) given that their mean mass number is \( M + 0.5 \).

### Step-by-Step Solution:

1. **Understanding the Mean Mass Number**:
The mean mass number of the isotopes can be calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of (mass number × ratio)}}{\text{Total ratio}} \]
Here, the isotopes have mass numbers \( M \), \( M + 1 \), and \( M + 2 \).

2. **Assigning Ratios**:
Let's denote the ratios of the isotopes as \( x \), \( y \), and \( z \) respectively. The mean mass number is given as \( M + 0.5 \). Thus, we can express the mean as:
\[ \text{Mean} = \frac{M \cdot x + (M + 1) \cdot y + (M + 2) \cdot z}{x + y + z} \]
Setting this equal to \( M + 0.5 \):
\[ \frac{M \cdot x + (M + 1) \cdot y + (M + 2) \cdot z}{x + y + z} = M + 0.5 \]

3. **Simplifying the Equation**:
Multiplying both sides by \( x + y + z \):
\[ M \cdot x + (M + 1) \cdot y + (M + 2) \cdot z = (M + 0.5)(x + y + z) \]
Expanding both sides:
\[ Mx + My + y + Mz + 2z = Mx + My + Mz + 0.5x + 0.5y + 0.5z \]

4. **Rearranging Terms**:
Canceling \( Mx + My + Mz \) from both sides:
\[ y + 2z = 0.5x + 0.5y + 0.5z \]
Rearranging gives:
\[ 0.5y + 1.5z = 0.5x \]
Multiplying through by 2 to eliminate the fractions:
\[ y + 3z = x \]

5. **Finding Ratios**:
We can express \( x \) in terms of \( y \) and \( z \):
\[ x = y + 3z \]
This means the ratios \( x : y : z \) can be expressed as:
\[ (y + 3z) : y : z \]
Let’s assume \( z = 1 \):
\[ x = y + 3 \quad \text{and} \quad y = y \]
Therefore, the ratios become:
\[ (y + 3) : y : 1 \]

6. **Testing Possible Ratios**:
We can test various integer values for \( y \) to find acceptable ratios:
- If \( y = 1 \), then \( x = 4 \), giving the ratio \( 4 : 1 : 1 \).
- If \( y = 2 \), then \( x = 5 \), giving the ratio \( 5 : 2 : 1 \).
- If \( y = 3 \), then \( x = 6 \), giving the ratio \( 6 : 3 : 1 \).

7. **Conclusion**:
Among the options provided, the ratio \( 4 : 1 : 1 \) satisfies the condition that the mean mass number is \( M + 0.5 \).

### Final Answer:
The accepted ratio for \( M \), \( M + 1 \), and \( M + 2 \) is \( 4 : 1 : 1 \).