If the arithmetic means of two positive number a and b `(a gt b )` is twice their geometric mean, then find the ratio a: b

Explanation

To solve the problem, we need to find the ratio \( a : b \) given that the arithmetic mean of two positive numbers \( a \) and \( b \) (where \( a > b \)) is twice their geometric mean.

### Step-by-Step Solution:

1. **Define the Arithmetic Mean (AM) and Geometric Mean (GM)**:
- The arithmetic mean of \( a \) and \( b \) is given by:
\[ AM = \frac{a + b}{2} \]
- The geometric mean of \( a \) and \( b \) is given by:
\[ GM = \sqrt{ab} \]

2. **Set Up the Equation**:
- According to the problem, the arithmetic mean is twice the geometric mean:
\[ \frac{a + b}{2} = 2 \sqrt{ab} \]

3. **Multiply Both Sides by 2**:
- To eliminate the fraction, multiply both sides of the equation by 2:
\[ a + b = 4 \sqrt{ab} \]

4. **Rearrange the Equation**:
- Rearranging gives us:
\[ a + b - 4 \sqrt{ab} = 0 \]

5. **Use Substitution**:
- Let \( \sqrt{a} = x \) and \( \sqrt{b} = y \). Then \( a = x^2 \) and \( b = y^2 \).
- Substitute into the equation:
\[ x^2 + y^2 - 4xy = 0 \]

6. **Rearrange the Substituted Equation**:
- Rearranging gives:
\[ x^2 + y^2 = 4xy \]

7. **Rearranging Further**:
- This can be rearranged to:
\[ x^2 - 4xy + y^2 = 0 \]

8. **Factor the Quadratic**:
- This is a quadratic in terms of \( x \):
\[ (x - 2y)^2 = 0 \]
- Therefore, we have:
\[ x - 2y = 0 \quad \Rightarrow \quad x = 2y \]

9. **Substitute Back to Find the Ratio**:
- Recall that \( x = \sqrt{a} \) and \( y = \sqrt{b} \):
\[ \sqrt{a} = 2\sqrt{b} \]
- Squaring both sides gives:
\[ a = 4b \]

10. **Find the Ratio \( a : b \)**:
- Thus, the ratio \( a : b \) is:
\[ a : b = 4 : 1 \]

### Final Answer:
The ratio \( a : b \) is \( 4 : 1 \).