Math Problem Statement
The AM of two observations is 25, and GM is 15. Find the two observations and their HM.
Solution
Given:
- Arithmetic Mean (AM) of two observations = 25
- Geometric Mean (GM) of two observations = 15
Let the two observations be and .
Step 1: Use the AM to form the first equation
The AM of and is given by:
Multiplying both sides by 2:
Step 2: Use the GM to form the second equation
The GM of and is given by:
Squaring both sides:
Step 3: Solve the system of equations
We now have two equations:
These equations can be solved using the quadratic formula. Consider the quadratic equation:
Substituting the values from Equations 1 and 2:
Step 4: Solve the quadratic equation
To solve this, use the quadratic formula:
For our equation :
So, the two roots (observations) are:
Therefore, the two observations are and .
Step 5: Calculate the Harmonic Mean (HM)
The Harmonic Mean (HM) of and is given by:
Substituting the values:
Final Answer
- The two observations are 45 and 5.
- The Harmonic Mean (HM) is 9.
Would you like any more details on this solution, or do you have any further questions?
Here are 5 related questions you might find interesting:
- How does the AM-GM-HM inequality relate to the values found here?
- What are the conditions under which the AM equals the GM or HM?
- How would the solution change if the GM were different?
- How can you use these means to estimate the range of two observations?
- What is the relationship between the AM, GM, and HM in a set of more than two observations?
Tip: The Harmonic Mean is most useful when averaging rates or ratios, particularly when the quantities have the same units but are distributed unevenly.
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Math Problem Analysis
Mathematical Concepts
Arithmetic Mean
Geometric Mean
Harmonic Mean
Quadratic Equations
Formulas
Arithmetic Mean formula
Geometric Mean formula
Harmonic Mean formula
Quadratic formula
Theorems
AM-GM inequality
Suitable Grade Level
Grades 9-12