The AM of two observations is 25, and GM is 15. Find the two observations and their HM.Without using quadratic equation

Given:
- Arithmetic Mean (AM) of two observations \( a \) and \( b \) is 25.
- Geometric Mean (GM) of \( a \) and \( b \) is 15.

We are to find the two observations \( a \) and \( b \), and their Harmonic Mean (HM).

### Step 1: Use the Arithmetic Mean (AM) formula
The formula for the AM of two numbers \( a \) and \( b \) is:

\[
\text{AM} = \frac{a + b}{2}
\]

Given \( \text{AM} = 25 \):

\[
\frac{a + b}{2} = 25
\]

Multiplying both sides by 2:

\[
a + b = 50
\]

### Step 2: Use the Geometric Mean (GM) formula
The formula for the GM of two numbers \( a \) and \( b \) is:

\[
\text{GM} = \sqrt{ab}
\]

Given \( \text{GM} = 15 \):

\[
\sqrt{ab} = 15
\]

Squaring both sides:

\[
ab = 225
\]

### Step 3: Find the two observations \( a \) and \( b \)
We now have the system of equations:

\[
a + b = 50
\]
\[
ab = 225
\]

To find \( a \) and \( b \), observe that these are the roots of the quadratic equation \( x^2 - (a+b)x + ab = 0 \). However, since we are not using the quadratic formula, we proceed by solving it directly.

Assume \( a \) and \( b \) are the roots of the equation:

\[
x^2 - 50x + 225 = 0
\]

We can factor this by finding two numbers that multiply to 225 and add to 50. These numbers are 25 and 9.

Thus, the two observations are:
\[
a = 25 \quad \text{and} \quad b = 9
\]

### Step 4: Calculate the Harmonic Mean (HM)
The formula for the Harmonic Mean (HM) of two numbers \( a \) and \( b \) is:

\[
\text{HM} = \frac{2ab}{a + b}
\]

Substituting the values of \( a = 25 \) and \( b = 9 \):

\[
\text{HM} = \frac{2 \times 25 \times 9}{25 + 9} = \frac{450}{34} \approx 13.24
\]

### Final Answer
The two observations are 25 and 9, and their Harmonic Mean is approximately \( 13.24 \).

Would you like more details or have any questions?

### Related Questions
1. How do you derive the formula for the Harmonic Mean?
2. Can you find the observations if the GM was 20 instead of 15?
3. What is the relationship between AM, GM, and HM for any two positive numbers?
4. How would the solution change if the AM was 30 and the GM was 18?
5. Can you demonstrate the same problem using a graphical method?

**Tip:** When working with means (AM, GM, HM), remember that AM ≥ GM ≥ HM for any set of positive numbers.

Explore how to find two observations given their Arithmetic Mean (AM) and Geometric Mean (GM) without using the quadratic equation. This problem involves calculating the AM, GM, and HM of two numbers, suitable for advanced high school students.

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