Question

If the geometric mean of two numbers is 6.0 and the arithmetic mean is 6.5, then the difference of squares of these numbers is 
1. 65
2. 120
3. 130
4. 140

Answer 1

Correct Answer - Option 1 : 65

Concept:

Let the two numbers as a and b.

Arithmetic Mean = \(\rm \dfrac {a+b}{2}\)

Geometric Mean = \(\rm√{ab}\) 

The relation between AM and GM:

(Geometric Mean)2 = (Arithmetic mean)(Harmonic mean)

Calculation:

Let the numbers be a and b respectively

(a + b)/2 = 6.5

∴ a + b = 13  ----(1)

√(ab) = 6

∴ ab = 36

We know that;

\((a-b)=\sqrt{(a+b)^2-4ab}\)

from equation 1) and 2)

= \(\sqrt{(13^2 - 4 × 36)} =\sqrt{169-144}\)

= √(25) = 5

∴ a - b = 5  ---- (2)

Adding equations 1 and 2

2a = 18

∴ a = 9 and b = 13 - 9 = 4

∴ The numbers are 16 and 4

The difference of squares of these numbers is:

= (9)² - (4)²

= 81 - 16

= 65