If the AM and HM of two numbers are 27 and 12 respectively, then what is their GM equal to ?

Options

A. 12
B. 18
C. 24
D. 27

Explanation

To find the geometric mean (GM) of two numbers given their arithmetic mean (AM) and harmonic mean (HM), we can use the relationship between these means.

### Step-by-step Solution:

1. **Identify the Given Values**:
- Arithmetic Mean (AM) = 27
- Harmonic Mean (HM) = 12

2. **Use the Relationship Between Means**:
The relationship between the arithmetic mean, harmonic mean, and geometric mean (GM) is given by the formula:
\[ GM^2 = AM \times HM \]

3. **Substitute the Known Values**:
Substitute the values of AM and HM into the formula:
\[ GM^2 = 27 \times 12 \]

4. **Calculate the Product**:
Calculate the product:
\[ GM^2 = 324 \]

5. **Find the Geometric Mean**:
To find GM, take the square root of both sides:
\[ GM = \sqrt{324} \]
\[ GM = 18 \]

### Final Answer:
The geometric mean (GM) of the two numbers is **18**.