Step-by-step Solution:

The Arithmetic Mean (AM) of two numbers \( a \) and \( b \) is defined as: \[ \frac{a + b}{2} = 5 \implies a + b = 10 \] The Harmonic Mean (HM) of two numbers \( a \) and \( b \) is given by: \[ \frac{2ab}{a + b} = 3.2 \] Substitute the derived sum \( a + b = 10 \) into the HM equation: \[ \frac{2ab}{10} = 3.2 \] \[ 2ab = 32 \implies ab = 16 \] We are looking for two positive numbers whose sum is 10 and whose product is 16. These numbers satisfy the following quadratic equation: \[ x^2 - (a+b)x + ab = 0 \] \[ x^2 - 10x + 16 = 0 \] Factoring the quadratic equation: \[ (x - 2)(x - 8) = 0 \] The roots are 2 and 8. Thus, the corresponding numbers are \( a=2 \) and \( b=8 \) (or vice versa).