Let `alpha` and `beta` be two number where `alpha lt beta` The geometric mean of these numbers exceeds the smaller `alpha` by 12 and the arithmetic mean of the same number is smaller by 24 than the larger number `beta` , then the value of `|beta-alpha|` is

Answer

To solve the problem, we will use the definitions of the geometric mean and arithmetic mean, and set up equations based on the information provided.

### Step-by-Step Solution:

1. **Define the Geometric Mean**:
The geometric mean of two numbers \( \alpha \) and \( \beta \) is given by:
\[ GM = \sqrt{\alpha \beta} \]
According to the problem, the geometric mean exceeds the smaller number \( \alpha \) by 12:
\[ \sqrt{\alpha \beta} = \alpha + 12 \]

2. **Define the Arithmetic Mean**:
The arithmetic mean of \( \alpha \) and \( \beta \) is given by:
\[ AM = \frac{\alpha + \beta}{2} \]
The problem states that this arithmetic mean is smaller by 24 than the larger number \( \beta \):
\[ \frac{\alpha + \beta}{2} = \beta - 24 \]

3. **Set up the equations**:
From the geometric mean equation:
\[ \sqrt{\alpha \beta} = \alpha + 12 \quad \text{(1)} \]
From the arithmetic mean equation, we can multiply both sides by 2:
\[ \alpha + \beta = 2\beta - 48 \quad \text{(2)} \]

4. **Rearranging Equation (2)**:
Rearranging equation (2) gives:
\[ \alpha + \beta - 2\beta = -48 \]
Simplifying this, we have:
\[ \alpha - \beta = -48 \quad \Rightarrow \quad \beta - \alpha = 48 \quad \text{(3)} \]

5. **Finding the Value of \( | \beta - \alpha | \)**:
From equation (3), we find:
\[ | \beta - \alpha | = 48 \]

Thus, the value of \( | \beta - \alpha | \) is \( 48 \).