Input the harmonic mean and arithmetic mean of two numbers to easily find their exact values.
Let the two numbers be \(x\) and \(y\), with their harmonic mean (\(H\)) and arithmetic mean (\(A\)) provided.
Harmonic Mean (H): The reciprocal of the average of the reciprocals of two numbers: \( H = \frac{2xy}{x + y} \)
Arithmetic Mean (A): The average of two numbers: \( A = \frac{x + y}{2} \)
Calculate the Sum (\(x + y\)) from the arithmetic mean: \( x + y = 2A \) Calculate the Product (\(xy\)) from the harmonic mean: \( \frac{2xy}{x + y} = H \) \( \frac{2xy}{2A} = H \) Simplify: \( xy = H \cdot A \) Use the sum and product to construct the quadratic equation: \( t^2 - (x + y)t + xy = 0 \) Substitute the values of \(x + y\) and \(xy\): \( t^2 - 2At + H \cdot A = 0 \) Using the quadratic formula to solve the quadratic equation: \( t = \frac{2A \pm \sqrt{(2A)^2 - 4 \cdot H \cdot A}}{2} \) Simplify: \( t = A \pm \sqrt{A^2 - H \cdot A} \) The two solutions, \(t = x\) and \(t = y\), provide the values of the numbers.
Solution:
1. Calculate the Sum \( x + y \):
\( x + y = 2 \cdot 10 = 20 \)
2. Calculate the Product \( x \cdot y \):
\( xy = 6.4 \cdot 10 = 64 \)
3. Form the Quadratic Equation:
\( t^2 - 20t + 64 = 0 \)
4. Solve the Equation:
\( t = \frac{20 \pm \sqrt{20^2 - 4 \cdot 64}}{2} = \frac{20 \pm \sqrt{400 - 256}}{2} = \frac{20 \pm \sqrt{144}}{2} \)
\( t_1 = \frac{20 + \sqrt{144}}{2} = 16 \)
\( t_2 = \frac{20 - \sqrt{144}}{2} = 4 \)
Result: The two numbers are 16 and 4.
Solution:
1. Calculate the Sum \( x + y \):
\( x + y = 2 \cdot 4 = 8 \)
2. Calculate the Product \( x \cdot y \):
\( xy = 3.75 \cdot 4 = 15 \)
3. Form the Quadratic Equation:
\( t^2 - 8t + 15 = 0 \)
4. Solve the Equation:
\( t = \frac{8 \pm \sqrt{8^2 - 4 \cdot 15}}{2} = \frac{8 \pm \sqrt{64 - 60}}{2} = \frac{8 \pm \sqrt{4}}{2} \)
\( t_1 = \frac{8 + \sqrt{4}}{2} = 5 \)
\( t_2 = \frac{8 - \sqrt{4}}{2} = 3 \)
Result: The two numbers are 5 and 3.