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

Answer

To solve the problem where the ratio of the Arithmetic Mean (AM) to the Geometric Mean (GM) of two positive numbers \( a \) and \( b \) is given as \( 5:3 \), we can follow these steps:

### Step 1: Understand the relationship between 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:
\[ GM = \sqrt{ab} \]

### Step 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:
\[ \frac{\frac{a + b}{2}}{\sqrt{ab}} = \frac{5}{3} \]

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

### Step 4: Square both sides to eliminate the square root
Squaring both sides results in:
\[ (3(a + b))^2 = (10\sqrt{ab})^2 \]
This simplifies to:
\[ 9(a + b)^2 = 100ab \]

### Step 5: Expand and rearrange the equation
Expanding the left side:
\[ 9(a^2 + 2ab + b^2) = 100ab \]
Rearranging gives:
\[ 9a^2 + 18ab + 9b^2 - 100ab = 0 \]
This simplifies to:
\[ 9a^2 - 82ab + 9b^2 = 0 \]

### Step 6: Use the quadratic formula
This is a quadratic equation in terms of \( a \):
\[ 9a^2 - 82ab + 9b^2 = 0 \]
Using the quadratic formula \( a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 9 \), \( B = -82b \), and \( C = 9b^2 \):
\[ a = \frac{82b \pm \sqrt{(-82b)^2 - 4 \cdot 9 \cdot 9b^2}}{2 \cdot 9} \]
Calculating the discriminant:
\[ (-82b)^2 - 4 \cdot 9 \cdot 9b^2 = 6724b^2 - 324b^2 = 6400b^2 \]
Thus:
\[ a = \frac{82b \pm 80b}{18} \]

### Step 7: Find the values of \( a \)
Calculating the two possible values:
1. \( a = \frac{162b}{18} = 9b \)
2. \( a = \frac{2b}{18} = \frac{b}{9} \)

### Step 8: Determine the ratio \( a:b \)
The two ratios we found are:
1. \( a:b = 9:1 \)
2. \( a:b = \frac{1}{9}:1 \) (which is not valid since \( a \) must be positive)

Thus, the valid ratio is:
\[ a:b = 9:1 \]

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

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