Question

The arithmetic mean and the harmonic mean between 2 numbers are 27 and 12 respectively, then their geometric mean is given by:

• A. 15
• B. 18
• C. 17
• D. 16

Answer 1

The Correct Option is B

Let's use the formulas for the arithmetic mean (AM), harmonic mean (HM), and geometric mean (GM) for two numbers a and b
\(1. ( AM = \frac{a + b}{2} )\)
\(2. ( HM = \frac{2}{\frac{1}{a} + \frac{1}{b}} )\)
\(3. ( GM = \sqrt{a \times b} )\)

Given:
AM = 27
HM = 12

From the given AM:
\(( \frac{a + b}{2} = 27 )\)
\(( a + b = 54 ) .......(i)\)

From the given HM:
\(\frac{2}{\frac{1}{a} + \frac{1}{b}} = 12\)

\(\frac{2ab}{a + b} = 12\)

\(2ab = 12(a + b)\)
\(2ab = 12(54)\)
\(2ab = 648\)
\([ ab = 324 ] .......(ii)\)

Now, the geometric mean is
\(GM = \sqrt{a \times b}\)
\(GM = \sqrt{324}\)
\(GM = 18\)

So, the correct answer is B : 18.