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

Options

A. (a) `3:5`
B. (b) `2:9`
C. (c) `9:1`
D. (d) `5:3`

Answer

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

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

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

4. **Rearranging the equation**:
- We can rewrite this as:
\[ 3a + 3b = 10\sqrt{ab} \]

5. **Square both sides to eliminate the square root**:
- Squaring both sides results in:
\[ (3a + 3b)^2 = (10\sqrt{ab})^2 \]
- This simplifies to:
\[ 9(a + b)^2 = 100ab \]

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

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

8. **Factoring the quadratic equation**:
- This can be factored as:
\[ 9a^2 - 81ab - 1ab + 9b^2 = 0 \]
- Grouping terms:
\[ 9a(a - 9b) - b(a - 9b) = 0 \]
- Factoring out \( (a - 9b) \):
\[ (a - 9b)(9a - b) = 0 \]

9. **Finding the solutions**:
- Setting each factor to zero gives:
- \( a - 9b = 0 \) which implies \( a = 9b \)
- \( 9a - b = 0 \) which implies \( b = 9a \)

10. **Finding the ratio \( a:b \)**:
- From \( a = 9b \), the ratio \( a:b \) is:
\[ \frac{a}{b} = \frac{9b}{b} = 9:1 \]
- From \( b = 9a \), the ratio \( a:b \) is:
\[ \frac{a}{b} = \frac{a}{9a} = 1:9 \]

11. **Conclusion**:
- The valid ratio from the options provided is \( 9:1 \).

Thus, the final answer is:
\[ \text{The ratio } a:b \text{ is } 9:1. \]