AM and GM of two numbers are 5 and 4 respectively find the two numbers

Answer (Detailed Solution Below)

Option 2 : 6 and 3

Free

120 Qs. 480 Marks 120 Mins

Concept:

Let x and y be the two numbers. The the arithmetic mean A,  geometric mean G and the harmonic mean H of x and y is given by, 

⇒ A =  

⇒ G2 = xy

⇒  

Calculations:

Consider, the two numbers are x and y.

Given, the arithmetic mean and geometric mean of the x and y is A and G.

⇒ A =            ....(1)

⇒ G2 = xy               ....(2)

The harmonic mean of two number x and y is 4.

⇒  

⇒ 2xy = 4(x + y)

⇒  

⇒ G2 = 4A                   (∵ x + y = 2A)

⇒ G2 = 4A            ....(3)

Given, Their arithmetic mean A and the geometric mean G satisfy the relation 2A + G2 = 27.

⇒2A + G2 = 27

⇒ 6A = 27

⇒ A = 

From equation (1), (2) and (3), we have

 x + y = 9 and xy = 18

⇒ x = 6 and y = 3

Hence, the harmonic mean of two number is 4, Their arithmetic mean A and the geometric mean G satisfy the relation 2A + G2 = 27, then the two numbers are 6 and 3.

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If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.