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 will start by using the definitions of arithmetic mean (AM) and geometric mean (GM) for two positive numbers \( a \) and \( b \) where \( a > b \).

### Step-by-Step Solution:

1. **Define Arithmetic Mean (AM)**:
The arithmetic mean of two numbers \( a \) and \( b \) is given by:
\[ AM = \frac{a + b}{2} \]

2. **Define Geometric Mean (GM)**:
The geometric mean of two numbers \( a \) and \( b \) is given by:
\[ GM = \sqrt{ab} \]

3. **Given Condition**:
According to the problem, the arithmetic mean is twice the geometric mean:
\[ AM = 2 \times GM \]
Substituting the expressions for AM and GM, we get:
\[ \frac{a + b}{2} = 2 \sqrt{ab} \]

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

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

6. **Using the Quadratic Formula**:
We can treat this as a quadratic equation in terms of \( a \) and \( b \). Let's express \( a \) in terms of \( b \):
\[ a = 4 \sqrt{ab} - b \]
Now, substituting \( a \) into the equation:
\[ (4 \sqrt{ab} - b) + b - 4 \sqrt{ab} = 0 \]
This simplifies to:
\[ 4 \sqrt{ab} = b \]

7. **Squaring Both Sides**:
Squaring both sides gives:
\[ 16ab = b^2 \]

8. **Rearranging**:
Rearranging this gives:
\[ b^2 - 16ab = 0 \]
Factoring out \( b \):
\[ b(b - 16a) = 0 \]

9. **Finding the Ratio**:
Since \( b \neq 0 \), we have:
\[ b = 16a \]
Therefore, the ratio \( \frac{a}{b} \) is:
\[ \frac{a}{b} = \frac{a}{16a} = \frac{1}{16} \]
Thus, the ratio \( a:b \) is:
\[ a:b = 1:16 \]

### Final Answer:
The ratio \( a:b \) is \( 1:16 \).