The harmonic mean and the geometric mean of two numbers are 10 and 12 respectively. What is their arithmetic mean?
14.4
Understanding the Relationship Between Means
This question asks us to find the arithmetic mean of two numbers when their harmonic mean and geometric mean are known. There's a specific relationship that connects the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H) for any two positive numbers. This relationship is given by the formula: \(G^2 = AH\).
In this problem, we are given the following values:
- Harmonic Mean (H) = 10
- Geometric Mean (G) = 12
We need to find the Arithmetic Mean (A).
Calculating the Arithmetic Mean
We can use the relationship \(G^2 = AH\) to find the arithmetic mean. Let's substitute the given values into the formula:
\(\qquad G^2 = AH\)
Substitute G = 12 and H = 10:
\(\qquad 12^2 = A \times 10\)
Calculate the square of the geometric mean:
\(\qquad 144 = 10A\)
Now, we need to solve for A. Divide both sides of the equation by 10:
\(\qquad A = \frac{144}{10}\)
Perform the division:
\(\qquad A = 14.4\)
So, the arithmetic mean of the two numbers is 14.4.
Let's summarize the calculation steps:
| Step | Description | Calculation |
|---|---|---|
| 1 | Identify the given values | H = 10, G = 12 |
| 2 | Use the formula relating the means | \(G^2 = AH\) |
| 3 | Substitute known values | \(12^2 = A \times 10\) |
| 4 | Simplify the equation | \(144 = 10A\) |
| 5 | Solve for A | \(A = \frac{144}{10} = 14.4\) |
The calculated arithmetic mean is 14.4.
Revision Table: Key Concepts
| Mean | Definition (for two numbers a and b) | Relationship |
|---|---|---|
| Arithmetic Mean (A) | \(\frac{a+b}{2}\) | \(G^2 = AH\) (for positive numbers) |
| Geometric Mean (G) | \(\sqrt{ab}\) | |
| Harmonic Mean (H) | \(\frac{2}{\frac{1}{a}+\frac{1}{b}} = \frac{2ab}{a+b}\) |
Additional Information on AM, GM, and HM
For any set of positive numbers, there is an important inequality relating the three means: \(A \ge G \ge H\). Equality holds only when all the numbers are equal.
In this problem, we found A = 14.4, G = 12, and H = 10. Let's check if the inequality holds: \(14.4 \ge 12 \ge 10\). This is true, which gives us confidence in our calculated arithmetic mean.
The relationship \(G^2 = AH\) is a specific case of this inequality and is very useful when dealing with two numbers.
Important Questions from Relations between AM, GM, HM
If the product of n positive numbers is unity, then their sum is?
If p = tan2 x + cot2 x, then which one of the following is correct?
If \(a_1, a_2, a_3,...,a_n\) are positive real numbers whose product is a fixed number C, then the minimum value of \(a_1+a_2+...+a_n\) is
In an acute angled ΔABC, the least value of sec A + sec B + sec C is:
Let x be the HM and y be the GM of two positive numbers m and n. If 5x = 4y, then which one of the following is correct?
