Question

The arithmetic mean of two numbers is
6. If the geometric mean is also 6, then
find the numbers.
A.
10,8
B.
10,6
C.
6,6
D.
9,2​

Answer 1

Step-by-step explanation:

The arithmetic mean of two numbers is

6. If the geometric mean is also 6, then

find the numbers.

A.

10,8

Answer 2

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:Let \: 2 \: numbers \: be \: x \: and \: y.$

According to statement,

It is given that arithmetic mean between 2 numbers is 6.

We know,

$\rm :\longmapsto\:Arithmetic \: mean \: between \: two \: numbers \: x \: and \: y \: is$

$\rm :\longmapsto\:Arithmetic \: mean = \dfrac{x + y}{2}$

$\rm :\longmapsto\:6 = \dfrac{x + y}{2}$

$\rm :\longmapsto\:x + y = 12$

$\rm :\longmapsto\:y = 12 - x - - - (1)$

According to statement again,

Geometric mean between 2 numbers is 6.

We know,

$\rm :\longmapsto\:Geometric \: mean \: between \: 2 \: numbers \: x \: and \: y \: is$

$\rm :\longmapsto\:Geometric \: mean = \sqrt{xy}$

$\rm :\longmapsto\:6 = \sqrt{xy}$

$\rm :\longmapsto\:xy = 36$

$\rm :\longmapsto\:x(12 - x) = 36 \: \: \: \: \: \: \: \: \{ \: using \: (1) \}$

$\rm :\longmapsto\:12x - {x}^{2} = 36$

$\rm :\longmapsto\: {x}^{2} - 12x + 36 = 0$

$\rm :\longmapsto\: {x}^{2} - 6x - 6x+ 36 = 0$

$\rm :\longmapsto\:x(x - 6) - 6(x - 6) = 0$

$\rm :\longmapsto\:(x - 6)(x - 6) = 0$

$\bf\implies \:x = 6$

On substituting the value of x in equation (1), we get

$\bf\implies \:y = 6$

Hence,

• Numbers are 6 and 6

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf{ \: Option \: (c) \: is \: correct}}}$

Additional Information :-

1. The relationship between Arithmetic mean and Geometric mean is

$\rm :\longmapsto\:Arithmetic \: mean \geqslant Geometric \: mean$

2. If all the observations are equal, then

$\rm :\longmapsto\:Arithmetic \: mean = Geometric \: mean$

3. If n AM are inserted between two positive numbers a and b, the sum of AM is equal to n times the single AM between a and b.