The geometric mean and harmonic mean of two non negative observations are 10 and 8 respectively. Then what is the arithmetic mean of the observations equal to?

Options

A. 4
B. 9
C. 12,5
D. 2

Explanation

To solve the problem, we need to find the arithmetic mean of two non-negative observations given their geometric mean (GM) and harmonic mean (HM).

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Geometric Mean (GM) = 10
- Harmonic Mean (HM) = 8

2. **Formulate the Equations**:
- Let the two observations be \( a \) and \( b \).
- The formula for the geometric mean of two numbers is:
\[ GM = \sqrt{ab} \]
Therefore, we have:
\[ \sqrt{ab} = 10 \implies ab = 10^2 = 100 \quad \text{(Equation 1)} \]

3. **Use the Harmonic Mean Formula**:
- The formula for the harmonic mean of two numbers is:
\[ HM = \frac{2ab}{a+b} \]
Given that HM = 8, we can write:
\[ \frac{2ab}{a+b} = 8 \]
Substituting \( ab = 100 \) from Equation 1:
\[ \frac{2 \times 100}{a+b} = 8 \]
Simplifying this gives:
\[ \frac{200}{a+b} = 8 \]
Rearranging yields:
\[ a+b = \frac{200}{8} = 25 \quad \text{(Equation 2)} \]

4. **Use the Algebraic Identity**:
- We can use the identity:
\[ (a-b)^2 = (a+b)^2 - 4ab \]
- Substituting the values from Equations 1 and 2:
\[ (a-b)^2 = (25)^2 - 4 \times 100 \]
Simplifying:
\[ (a-b)^2 = 625 - 400 = 225 \]
Taking the square root:
\[ a-b = \sqrt{225} = 15 \quad \text{(Equation 3)} \]

5. **Solve for \( a \) and \( b \)**:
- Now we have two equations:
- \( a + b = 25 \) (from Equation 2)
- \( a - b = 15 \) (from Equation 3)
- Adding these two equations:
\[ (a+b) + (a-b) = 25 + 15 \]
This simplifies to:
\[ 2a = 40 \implies a = 20 \]
- Now substituting \( a = 20 \) back into \( a + b = 25 \):
\[ 20 + b = 25 \implies b = 5 \]

6. **Calculate the Arithmetic Mean**:
- The arithmetic mean (AM) is given by:
\[ AM = \frac{a+b}{2} = \frac{20 + 5}{2} = \frac{25}{2} = 12.5 \]

### Final Answer:
The arithmetic mean of the two observations is \( \boxed{12.5} \).