The arithmetic mean between two distinct positive numbers is twice the geometric mean between them. Find the ratio of greater to smaller. ​

Explanation

To solve the problem, we need to find the ratio of two distinct positive numbers \( a \) and \( b \) given that the arithmetic mean (AM) between them is twice the geometric mean (GM).

### 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 by 2:
\[ a + b = 4\sqrt{ab} \]

4. **Square Both Sides**:
- To eliminate the square root, square both sides:
\[ (a + b)^2 = (4\sqrt{ab})^2 \]
- This simplifies to:
\[ a^2 + 2ab + b^2 = 16ab \]

5. **Rearrange the Equation**:
- Rearranging gives:
\[ a^2 + b^2 - 14ab = 0 \]

6. **Use the Identity**:
- We can use the identity \( a^2 + b^2 = (a + b)^2 - 2ab \):
\[ (a + b)^2 - 2ab - 14ab = 0 \implies (a + b)^2 - 16ab = 0 \]
- Thus, we have:
\[ a^2 + b^2 = 14ab \]

7. **Divide by \( ab \)**:
- Dividing the entire equation by \( ab \):
\[ \frac{a^2}{ab} + \frac{b^2}{ab} = 14 \]
- This simplifies to:
\[ \frac{a}{b} + \frac{b}{a} = 14 \]

8. **Let \( x = \frac{a}{b} \)**:
- Then, we can rewrite the equation as:
\[ x + \frac{1}{x} = 14 \]

9. **Multiply by \( x \)**:
- Multiply through by \( x \) to eliminate the fraction:
\[ x^2 - 14x + 1 = 0 \]

10. **Use the Quadratic Formula**:
- To find \( x \), apply the quadratic formula:
\[ x = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
- This simplifies to:
\[ x = \frac{14 \pm \sqrt{196 - 4}}{2} = \frac{14 \pm \sqrt{192}}{2} \]
- Further simplifying gives:
\[ x = \frac{14 \pm 8\sqrt{3}}{2} = 7 \pm 4\sqrt{3} \]

11. **Select the Positive Ratio**:
- Since we need the ratio of the greater to the smaller number, we take the positive root:
\[ x = 7 + 4\sqrt{3} \]

### Conclusion:
The ratio of the greater number to the smaller number is \( 7 + 4\sqrt{3} \).