If the ratio of AM to GM of two positive numbers a and b is `5:3`, then a : b is equal to

Options

A. `3:5`
B. `2:9`
C. `9:1`
D. `5:3`

Explanation

To solve the problem, we need to find the ratio \( a : b \) given that the ratio of the Arithmetic Mean (AM) to the Geometric Mean (GM) of two positive numbers \( a \) and \( b \) is \( 5:3 \).

### Step-by-Step Solution:

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

2. **Set up the ratio**:
- According to the problem, we have:
\[ \frac{AM}{GM} = \frac{5}{3} \]
- Substituting the expressions for AM and GM, we get:
\[ \frac{\frac{a + b}{2}}{\sqrt{ab}} = \frac{5}{3} \]

3. **Cross-multiply to eliminate the fraction**:
- Cross-multiplying gives:
\[ 3(a + b) = 10\sqrt{ab} \]

4. **Square both sides to eliminate the square root**:
- Squaring both sides results in:
\[ 9(a + b)^2 = 100ab \]

5. **Expand the left side**:
- Expanding \( (a + b)^2 \):
\[ 9(a^2 + 2ab + b^2) = 100ab \]
- This simplifies to:
\[ 9a^2 + 18ab + 9b^2 = 100ab \]

6. **Rearrange the equation**:
- Rearranging gives:
\[ 9a^2 + 9b^2 - 82ab = 0 \]

7. **Factor the quadratic equation**:
- This can be factored as:
\[ 9a^2 - 82ab + 9b^2 = 0 \]

8. **Use the quadratic formula**:
- Using the quadratic formula \( a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 9, B = -82b, C = 9b^2 \):
\[ a = \frac{82b \pm \sqrt{(-82b)^2 - 4 \cdot 9 \cdot 9b^2}}{2 \cdot 9} \]

9. **Calculate the discriminant**:
- The discriminant simplifies to:
\[ 82^2b^2 - 324b^2 = (6724 - 324)b^2 = 6400b^2 \]
- Thus, the square root is:
\[ \sqrt{6400b^2} = 80b \]

10. **Substitute back into the formula**:
- Now substituting back gives:
\[ a = \frac{82b \pm 80b}{18} \]
- This results in two possible values:
\[ a = \frac{162b}{18} = 9b \quad \text{or} \quad a = \frac{2b}{18} = \frac{b}{9} \]

11. **Find the ratio \( a:b \)**:
- From \( a = 9b \), we have:
\[ a:b = 9:1 \]
- From \( a = \frac{b}{9} \), we have:
\[ a:b = 1:9 \]

12. **Conclusion**:
- The ratio \( a:b \) can be expressed as \( 9:1 \) or \( 1:9 \), depending on the values of \( a \) and \( b \).

### Final Answer:
The ratio \( a : b \) is \( 9 : 1 \).