Question

Arithmetic mean of two numbers \(a\) and \(b\) is \(5\) and the harmonic mean is \(3.2\). Find the numbers \(a\) and \(b\)?

• A. \(a=3,\; b=7\)
• B. \(a=4,\; b=6\)
• C. \(a=1,\; b=9\)
• D. \(a=2,\; b=8\)

Hint:

For two numbers, if A.M. and H.M. are known, first find \(a+b\) from the arithmetic mean and then find \(ab\) from the harmonic mean. The numbers can then be obtained using the quadratic equation \(x^2-(a+b)x+ab=0\).

Answer 1

The Correct Option is D

Concept: For two numbers \(a\) and \(b\), \[ \text{Arithmetic Mean (A.M.)}=\frac{a+b}{2} \] and \[ \text{Harmonic Mean (H.M.)}=\frac{2ab}{a+b} \] Using the given values of A.M. and H.M., we can first determine the sum of the numbers and then their product. Once the sum and product are known, the numbers can be obtained by forming a quadratic equation.

Step 1: Use the given Arithmetic Mean.
We are given that \[ \frac{a+b}{2}=5 \] Multiplying both sides by \(2\), \[ a+b=10 \] Thus, the sum of the two numbers is \[ a+b=10. \]

Step 2: Use the given Harmonic Mean.
The harmonic mean is \[ \frac{2ab}{a+b}=3.2 \] Substituting \(a+b=10\), \[ \frac{2ab}{10}=3.2 \] \[ \frac{ab}{5}=3.2 \] \[ ab=16 \] Hence, the product of the two numbers is \[ ab=16. \]

Step 3: Form the quadratic equation.
The numbers are roots of the equation \[ x^2-(a+b)x+ab=0 \] Substituting \(a+b=10\) and \(ab=16\), \[ x^2-10x+16=0 \] Factorizing, \[ x^2-10x+16=(x-2)(x-8) \] Therefore, \[ x=2 \quad \text{or} \quad x=8. \] Hence, the two numbers are \[ a=2,\qquad b=8. \]

Step 4: Verification of the answer.
Arithmetic Mean: \[ \frac{2+8}{2}=5 \] Harmonic Mean: \[ \frac{2(2)(8)}{2+8} = \frac{32}{10} = 3.2 \] Both conditions are satisfied. Therefore, the required numbers are \[ \boxed{2 \text{ and } 8} \]