The arithmetic mean of two numbers is 10 and their geometric mean is 8. What are the two numbers?
16, 4
Understanding Arithmetic Mean and Geometric Mean
The question asks us to find two numbers given their arithmetic mean (AM) and geometric mean (GM). Let the two numbers be denoted as \(a\) and \(b\).
The arithmetic mean of two numbers \(a\) and \(b\) is defined as:
\( \text{AM} = \frac{a+b}{2} \)
The geometric mean of two positive numbers \(a\) and \(b\) is defined as:
\( \text{GM} = \sqrt{ab} \)
Setting Up Equations from the Given Information
We are given that the arithmetic mean of the two numbers is 10. So, we can write the first equation:
\( \frac{a+b}{2} = 10 \)
Multiplying both sides by 2, we get:
\( a+b = 20 \quad (Equation \; 1) \)
We are also given that the geometric mean of the two numbers is 8. So, we can write the second equation:
\( \sqrt{ab} = 8 \)
Squaring both sides to eliminate the square root, we get:
\( ab = 8^2 \)
\( ab = 64 \quad (Equation \; 2) \)
Solving the System of Equations
We now have a system of two equations with two variables \(a\) and \(b\):
- \( a+b = 20 \)
- \( ab = 64 \)
We can solve this system. From Equation 1, we can express \(a\) in terms of \(b\) (or vice versa):
\( a = 20 - b \)
Substitute this expression for \(a\) into Equation 2:
\( (20 - b)b = 64 \)
Expand the equation:
\( 20b - b^2 = 64 \)
Rearrange the terms to form a standard quadratic equation:
\( b^2 - 20b + 64 = 0 \)
Now we need to solve this quadratic equation for \(b\). We can factor this quadratic equation. We need two numbers that multiply to 64 and add up to -20. These numbers are -16 and -4.
\( (b - 16)(b - 4) = 0 \)
This gives us two possible values for \(b\):
\( b - 16 = 0 \implies b = 16 \)
\( b - 4 = 0 \implies b = 4 \)
If \(b=16\), substitute this back into \( a = 20 - b \):
\( a = 20 - 16 = 4 \)
The two numbers are 4 and 16.
If \(b=4\), substitute this back into \( a = 20 - b \):
\( a = 20 - 4 = 16 \)
The two numbers are 16 and 4.
In either case, the two numbers are 16 and 4.
Verifying the Solution with Options
Alternatively, we can check each option provided to see which pair of numbers satisfies both the arithmetic mean and geometric mean conditions.
| Option | Numbers (a, b) | Arithmetic Mean \( \frac{a+b}{2} \) | Geometric Mean \( \sqrt{ab} \) | Matches AM=10? | Matches GM=8? | Correct? |
|---|---|---|---|---|---|---|
| 1 | 15, 5 | \( \frac{15+5}{2} = \frac{20}{2} = 10 \) | \( \sqrt{15 \times 5} = \sqrt{75} \) | Yes | No | No |
| 2 | 12, 8 | \( \frac{12+8}{2} = \frac{20}{2} = 10 \) | \( \sqrt{12 \times 8} = \sqrt{96} \) | Yes | No | No |
| 3 | 16, 4 | \( \frac{16+4}{2} = \frac{20}{2} = 10 \) | \( \sqrt{16 \times 4} = \sqrt{64} = 8 \) | Yes | Yes | Yes |
| 4 | 18, 2 | \( \frac{18+2}{2} = \frac{20}{2} = 10 \) | \( \sqrt{18 \times 2} = \sqrt{36} = 6 \) | Yes | No | No |
From the table, only the pair (16, 4) satisfies both conditions: AM = 10 and GM = 8.
Thus, the two numbers are 16 and 4.
Revision Table: Key Concepts for AM and GM
| Concept | Definition (for two numbers \(a, b\)) | Property |
|---|---|---|
| Arithmetic Mean (AM) | \( \frac{a+b}{2} \) | Represents the average value. |
| Geometric Mean (GM) | \( \sqrt{ab} \) (for \(a, b > 0\)) | Used for averages of ratios or growth rates. |
| AM-GM Inequality | \( \text{AM} \ge \text{GM} \) | For non-negative numbers, AM is always greater than or equal to GM. Equality holds only when the numbers are equal. |
Additional Information: Solving AM and GM Problems
Problems involving arithmetic mean and geometric mean often require setting up equations based on their definitions. For two numbers, the sum and product can be found directly from the given AM and GM values.
- If AM is given as \(A\), then \( \frac{a+b}{2} = A \implies a+b = 2A \).
- If GM is given as \(G\), then \( \sqrt{ab} = G \implies ab = G^2 \).
Once the sum (\(a+b\)) and product (\(ab\)) of the two numbers are known, the numbers themselves can be found by solving the quadratic equation \( x^2 - (a+b)x + ab = 0 \). The roots of this quadratic equation will be the two numbers.
In this specific problem, we had \( a+b = 20 \) and \( ab = 64 \), leading to the quadratic equation \( x^2 - 20x + 64 = 0 \), which yielded the numbers 16 and 4.
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