The AM of two observations is 25, and GM is 15. Find the two observations and their HM.

Given:

- Arithmetic Mean (AM) of two observations: \( \text{AM} = 25 \)
- Geometric Mean (GM) of two observations: \( \text{GM} = 15 \)

Let the two observations be \( x_1 \) and \( x_2 \).

### Step 1: Using the AM formula

The Arithmetic Mean (AM) is given by:
\[
\text{AM} = \frac{x_1 + x_2}{2}
\]
Substituting the given AM value:
\[
25 = \frac{x_1 + x_2}{2}
\]
\[
x_1 + x_2 = 50 \quad \text{(Equation 1)}
\]

### Step 2: Using the GM formula

The Geometric Mean (GM) is given by:
\[
\text{GM} = \sqrt{x_1 \times x_2}
\]
Substituting the given GM value:
\[
15 = \sqrt{x_1 \times x_2}
\]
\[
x_1 \times x_2 = 225 \quad \text{(Equation 2)}
\]

### Step 3: Solving the equations

Now we have two equations:
\[
x_1 + x_2 = 50 \quad \text{(Equation 1)}
\]
\[
x_1 \times x_2 = 225 \quad \text{(Equation 2)}
\]

These are the sum and product of the roots of a quadratic equation:


Discover how to find two observations given their Arithmetic Mean (AM) is 25 and their Geometric Mean (GM) is 15. This problem involves applying the Arithmetic Mean and Geometric Mean formulas, along with solving quadratic equations. Suitable for Grades 9-12 students.

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