Question

If the arithmetic mean between two positive numbers is \(10\) and the geometric mean is \(8\), then the numbers are:

• A. \(6,14\)
• B. \(8,12\)
• C. \(4,16\)
• D. \(2,18\)

Hint:

If A.M. and G.M. of two numbers are known, first find: \[ a+b \quad \text{and} \quad ab \] then form the quadratic equation.

Answer 1

The Correct Option is A

Concept: For two positive numbers \(a\) and \(b\): \[ \text{Arithmetic Mean (A.M.)} = \frac{a+b}{2} \] and \[ \text{Geometric Mean (G.M.)} = \sqrt{ab} \] Using these two equations together, we can determine the numbers uniquely.

Step 1: Use the arithmetic mean condition. Given: \[ \frac{a+b}{2}=10 \] Multiply both sides by \(2\): \[ a+b=20 \] \[ \boxed{a+b=20} \]

Step 2: Use the geometric mean condition. We are also given: \[ \sqrt{ab}=8 \] Squaring both sides: \[ ab=64 \] \[ \boxed{ab=64} \]

Step 3: Form the quadratic equation. The numbers satisfy: \[ x^2-(a+b)x+ab=0 \] Substituting values: \[ x^2-20x+64=0 \]

Step 4: Solve the quadratic equation. Factorizing: \[ x^2-20x+64=0 \] \[ (x-16)(x-4)=0 \] Thus, \[ x=16 \quad \text{or} \quad x=4 \] Therefore, the two numbers are: \[ 4 \quad \text{and} \quad 16 \] Hence, \[ \boxed{(C)\ 4,16} \] Note: Although the provided answer key states option (A), the correct mathematical answer is actually: \[ \boxed{(C)\ 4,16} \] because: \[ \frac{4+16}{2}=10 \] and \[ \sqrt{4\times16}=\sqrt{64}=8 \]