Explanation
To solve this problem, we use the standard formulas for the Arithmetic Mean (AM) and Harmonic Mean (HM) of two numbers a and b.
1. Using the Arithmetic Mean:
The formula for the arithmetic mean is:
2a+b=5
a+b=10— (Equation 1)
2. Using the Harmonic Mean:
The formula for the harmonic mean is:
a+b2ab=3.2
Substitute (a+b)=10 from Equation 1 into this formula:
102ab=3.2
2ab=32
ab=16— (Equation 2)
3. Finding the numbers:
We need two numbers that sum to 10 and multiply to 16. We can form a quadratic equation x2−(a+b)x+ab=0:
x2−10x+16=0
Factor the quadratic:
(x−2)(x−8)=0
So, the values are x=2 and x=8.
Thus, the numbers are a=2 and b=8.
Correct Option: (d)
