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

Answer: (B)

18

Answer 2

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.