Question

The arithmetic mean and geometric mean of two positive numbers are 10 and 10 respectively. Determine the numbers.​

Answer 1

Step-by-step explanation:

a+b/2=10

a+b=20

√ab=10

ab=100

(a-b)^2=(a+b)^2-4ab

=400-400=0

a-b=0

a=b

a=b=10

Answer 2

Step-by-step explanation:

Given :-

The arithmetic mean and geometric mean of two positive numbers are 10 and 10 respectively.

To find :-

Determine the numbers.?

Solution :-

Let the required two numbers be X and Y

We know that

The Arithmetic mean of a and b = (a+b)/2

The Arithmetic Mean of X and Y = (X+Y)/2

Given that

The Arithmetic Mean of the numbers = 10

=> (X+Y)/2 = 10

=> X+Y = 10×2

=> X+Y = 20 ---------(1)

And

The Geometric Mean of a and b = √ab

The Geometric Mean of X and Y = √(XY)

Given that

The Geomertic Mean of the two numbers = 10

=> √XY = 10

On squaring both sides then

=> (√XY)^2 = 10^2

=> XY = 100 ------------(2)

We know that

(a-b)^2 = (a+b)^2-4ab

(X-Y)^2 = (X+Y)^2-4XY

=> (X-Y)^2 = 20^2 - 4(100)

=> (X-Y)^2 = 400-400

=> (X-Y)^2 = 0

=> X-Y = 0

=> X = Y ---------(3)

From (1)

Y+Y = 20

=> 2Y = 20

=> Y = 20/2

=> Y = 10

=> X = 10

Therefore, X = 10 and Y = 10

Answer:-

The required two numbers are 10 and 10

Check:-

AM of 10 and 10 = (10+10)/2 = 20/2 = 10

GM of 10 and 10 = √(10×10)=√100=10

Used formulae:-

• The Arithmetic mean of a and b = (a+b)/2

• The Geometric Mean of a and b = √ab