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 = \(\rm \dfrac {x + y}{2}\) ⇒ G2 = xy ⇒ \(\rm H = \dfrac {2xy}{x+y}\) 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 = \(\rm \dfrac {x + y}{2}\) ....(1) ⇒ G2 = xy ....(2) The harmonic mean of two number x and y is 4. ⇒ \(\rm \dfrac {2xy}{x+y}= 4\) ⇒ 2xy = 4(x + y) ⇒ \(\rm xy = 2(x+y)\) ⇒ 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 = \(\rm \dfrac 9{2}\) 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.
Last updated at Dec. 12, 2016 by Teachoo
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