The arithmetic mean of two numbers exceeds their geometric mean by 2 and the geometric mean exceeds their harmonic mean by 1.6. what are the two numbers ?

Options

A. 16,4
B. 81,9
C. 256,16
D. 625,25

Answer

To solve the problem, we need to find two numbers based on the relationships between their arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM).

1. **Define the means:**
Let the two numbers be \( A \) and \( B \).
- The arithmetic mean (AM) is given by:
\[ AM = \frac{A + B}{2} \]
- The geometric mean (GM) is given by:
\[ GM = \sqrt{AB} \]
- The harmonic mean (HM) is given by:
\[ HM = \frac{2AB}{A + B} \]

2. **Set up the equations:**
According to the problem:
- The arithmetic mean exceeds the geometric mean by 2:
\[ AM = GM + 2 \]
- The geometric mean exceeds the harmonic mean by 1.6:
\[ GM = HM + 1.6 \]

3. **Substituting the means:**
Substitute the expressions for AM, GM, and HM into the equations:
- From the first equation:
\[ \frac{A + B}{2} = \sqrt{AB} + 2 \]
- From the second equation:
\[ \sqrt{AB} = \frac{2AB}{A + B} + 1.6 \]

4. **Rearranging the first equation:**
Multiply both sides of the first equation by 2:
\[ A + B = 2\sqrt{AB} + 4 \]

5. **Rearranging the second equation:**
Multiply both sides of the second equation by \( A + B \):
\[ \sqrt{AB}(A + B) = 2AB + 1.6(A + B) \]
Rearranging gives:
\[ \sqrt{AB}(A + B) - 1.6(A + B) = 2AB \]
Factor out \( A + B \):
\[ (A + B)(\sqrt{AB} - 1.6) = 2AB \]

6. **Substituting \( A + B \) from the first equation into the second:**
From \( A + B = 2\sqrt{AB} + 4 \):
Substitute into the second equation:
\[ (2\sqrt{AB} + 4)(\sqrt{AB} - 1.6) = 2AB \]

7. **Expanding and simplifying:**
Expanding the left side:
\[ 2\sqrt{AB} \cdot \sqrt{AB} - 3.2\sqrt{AB} + 4\sqrt{AB} - 6.4 = 2AB \]
This simplifies to:
\[ 2AB - 3.2\sqrt{AB} + 4\sqrt{AB} - 6.4 = 2AB \]
Cancel \( 2AB \) from both sides:
\[ 0 = -3.2\sqrt{AB} + 4\sqrt{AB} - 6.4 \]
Combine like terms:
\[ 0 = 0.8\sqrt{AB} - 6.4 \]
Rearranging gives:
\[ 0.8\sqrt{AB} = 6.4 \]
Dividing both sides by 0.8:
\[ \sqrt{AB} = 8 \]

8. **Finding \( AB \):**
Squaring both sides:
\[ AB = 64 \]

9. **Finding \( A + B \):**
Substitute \( \sqrt{AB} \) back into the equation for \( A + B \):
\[ A + B = 2(8) + 4 = 16 \]

10. **Solving the system of equations:**
Now we have:
- \( A + B = 16 \)
- \( AB = 64 \)

These are the roots of the quadratic equation:
\[ x^2 - (A + B)x + AB = 0 \]
Substituting the values:
\[ x^2 - 16x + 64 = 0 \]

11. **Factoring the quadratic:**
This factors to:
\[ (x - 8)(x - 8) = 0 \]
Thus, \( x = 8 \).

12. **Conclusion:**
The two numbers are \( A = 8 \) and \( B = 8 \).