Two numbers A and B are such that their GM is 20% lower than their AM. Find the ratio between the numbers.

Options

A. `3:2`
B. `4:1`
C. `2:1`
D. `3:1`

Answer

To find the ratio between the numbers A and B, given that their geometric mean (GM) is 20% lower than their arithmetic mean (AM), we can follow these steps:

### Step 1: Understand the relationship between GM and AM
We know that:
\[ \text{GM} = \sqrt{AB} \]
\[ \text{AM} = \frac{A + B}{2} \]

According to the problem, the GM is 20% lower than the AM:
\[ \text{GM} = 0.8 \times \text{AM} \]

### Step 2: Set up the equation
Substituting the expressions for GM and AM into the equation:
\[ \sqrt{AB} = 0.8 \times \frac{A + B}{2} \]

### Step 3: Simplify the equation
Rearranging gives:
\[ \sqrt{AB} = \frac{0.8(A + B)}{2} \]
\[ \sqrt{AB} = 0.4(A + B) \]

### Step 4: Square both sides
To eliminate the square root, we square both sides:
\[ AB = (0.4(A + B))^2 \]
\[ AB = 0.16(A + B)^2 \]

### Step 5: Expand the right side
Expanding the right side:
\[ AB = 0.16(A^2 + 2AB + B^2) \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ AB = 0.16A^2 + 0.32AB + 0.16B^2 \]
Bringing all terms to one side:
\[ AB - 0.32AB = 0.16A^2 + 0.16B^2 \]
\[ 0.68AB = 0.16A^2 + 0.16B^2 \]

### Step 7: Divide by AB
Dividing through by AB gives:
\[ 0.68 = 0.16 \left(\frac{A^2}{AB} + \frac{B^2}{AB}\right) \]
This simplifies to:
\[ 0.68 = 0.16 \left(\frac{A}{B} + \frac{B}{A}\right) \]

### Step 8: Let \( x = \frac{A}{B} \)
Let \( x = \frac{A}{B} \), then \( \frac{B}{A} = \frac{1}{x} \):
\[ 0.68 = 0.16 \left(x + \frac{1}{x}\right) \]

### Step 9: Multiply through by \( x \)
Multiplying through by \( x \) gives:
\[ 0.68x = 0.16(x^2 + 1) \]
Rearranging gives:
\[ 0.16x^2 - 0.68x + 0.16 = 0 \]

### Step 10: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 0.16, b = -0.68, c = 0.16 \):
\[ x = \frac{0.68 \pm \sqrt{(-0.68)^2 - 4 \cdot 0.16 \cdot 0.16}}{2 \cdot 0.16} \]
\[ x = \frac{0.68 \pm \sqrt{0.4624 - 0.1024}}{0.32} \]
\[ x = \frac{0.68 \pm \sqrt{0.36}}{0.32} \]
\[ x = \frac{0.68 \pm 0.6}{0.32} \]

### Step 11: Calculate the possible values of \( x \)
Calculating the two possible values:
1. \( x = \frac{1.28}{0.32} = 4 \)
2. \( x = \frac{0.08}{0.32} = 0.25 \)

### Step 12: Determine the ratio
Thus, the ratios \( \frac{A}{B} \) can be either \( 4 \) or \( \frac{1}{4} \). Therefore, the ratio \( A:B \) can be either \( 4:1 \) or \( 1:4 \).

### Final Answer
The ratio between the numbers A and B is \( 4:1 \) or \( 1:4 \).

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