The A.M. of two positive numbers is 15 and their G.M is 12. What is the smaller number ?

Options

A. 8
B. 12
C. 6
D. 24

Answer

To find the smaller number given that the arithmetic mean (A.M.) of two positive numbers is 15 and their geometric mean (G.M.) is 12, we can follow these steps:

### Step 1: Set Up the Equations
Let the two positive numbers be \( X \) and \( Y \).

From the definition of arithmetic mean:
\[ \frac{X + Y}{2} = 15 \]
This implies:
\[ X + Y = 30 \quad \text{(Equation 1)} \]

From the definition of geometric mean:
\[ \sqrt{XY} = 12 \]
This implies:
\[ XY = 144 \quad \text{(Equation 2)} \]

### Step 2: Substitute and Rearrange
From Equation 1, we can express \( X \) in terms of \( Y \):
\[ X = 30 - Y \quad \text{(Equation 3)} \]

### Step 3: Substitute Equation 3 into Equation 2
Now, substitute Equation 3 into Equation 2:
\[ (30 - Y)Y = 144 \]
Expanding this gives:
\[ 30Y - Y^2 = 144 \]
Rearranging it leads to:
\[ Y^2 - 30Y + 144 = 0 \quad \text{(Quadratic Equation)} \]

### Step 4: Solve the Quadratic Equation
We can solve the quadratic equation using the quadratic formula:
\[ Y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -30 \), and \( c = 144 \):
\[ Y = \frac{30 \pm \sqrt{(-30)^2 - 4 \cdot 1 \cdot 144}}{2 \cdot 1} \]
Calculating the discriminant:
\[ Y = \frac{30 \pm \sqrt{900 - 576}}{2} \]
\[ Y = \frac{30 \pm \sqrt{324}}{2} \]
\[ Y = \frac{30 \pm 18}{2} \]

### Step 5: Calculate the Roots
Now, we find the two possible values for \( Y \):
1. \( Y = \frac{30 + 18}{2} = \frac{48}{2} = 24 \)
2. \( Y = \frac{30 - 18}{2} = \frac{12}{2} = 6 \)

### Step 6: Find the Corresponding Values of \( X \)
Using Equation 3, we can find the corresponding values of \( X \):
1. If \( Y = 24 \), then \( X = 30 - 24 = 6 \)
2. If \( Y = 6 \), then \( X = 30 - 6 = 24 \)

### Conclusion
The two numbers are \( 6 \) and \( 24 \). The smaller number is:
\[ \boxed{6} \]