Question

If the arithmetic mean of two number is 15 and geometric mean is 9 find the number

Answer 1

Topic

Sequence and Series

Given

Arithmetic Mean = 15 and

Geometric mean = 9.

To Find

The two numbers for which given statement satisfy.

Concept

Let two numbers as 'a' and 'b'.

Arithmetic Mean = ( a + b ) / 2

Geometric Mean = √(ab)

Solution

It is given that,

Arithmetic Mean = 15

( a + b ) / 2 = 15

a + b = 2 × 15

a + b = 30

a = 30 - b

It is also given that,

Geometric Mean = 9

√(ab) = 9

Square both sides,

ab = 81

( 30 - b )( b ) = 81

30b - b² = 81

b² - 30b + 81 = 0

b² - 27b -3b + 81 = 0

b( b - 27 ) - 3( b -27 ) = 0

( b -27 )( b - 3 ) = 0

Case 1

b - 27 = 0

b = 27

then

a = 30 - b

a = 30 - 27

a = 3

Case 2

b - 3 = 0

b = 3

then

a = 30 - b

a = 30 - 3

a = 27

Answer

So, two required numbers are 3 and 27.