The arithmetic mean of two numbers is 4 and their harmonic mean is 15 4 find the numbers

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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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