Question

The arithmetic mean of two numbers is 10 and their geometric mean is 6 the difference between numbers is

Answer 1

let two numbers are a and b
AM = (a+b)/2 Equation 1
GM =√a*b
a+b = 20
√a*b =6
square both sides
ab = 36
b =36/a

put this value in 1st Equation
a + 36/a = 20
a^2 - 20*a + 36 =0
a = (20 + 16)/2
a=18
b =2
two numbers are 18 and 2

Answer 2

Answer:

The difference between the number is $18,2$

Step-by-step explanation:

Given: The arithmetic mean of two numbers is 10 and their geometric mean is 6

To find: The difference between the number

Solution:

Let the numbers be $a, b .$

So, arithmetic mean $=\frac{a+b}{2}$

$\begin{aligned}&\Rightarrow \frac{a+b}{2}=10 \\&\Rightarrow a+b=20\end{aligned}$

Geometric mean $=\sqrt{a b}$

$\begin{aligned}&\Rightarrow \sqrt{a b}=6 \\&\Rightarrow a b=36 \\&\text { Since } a+b=20 \Rightarrow b=20-a \\&\Rightarrow a \times(20-a)=36 \\&\Rightarrow a^{2}-20 a+36=0 \\&\Rightarrow a^{2}-18 a-2a+36=0 \\\&\Rightarrow a(a-18)-2(a-18)=0 \\\&\Rightarrow(a-18) \times(a-2)=0 \\&\Rightarrow a=18 \text { or } a=2 .\end{aligned}$

So, the numbers are $18, 2$.