Question

if the ratio of H.M & G.M of two quantities is 12:13 than show that the ratio of the numbers is 9:4​

Answer 1

Answer:

H.M. = 2ab/(a+b)

G.M. = (ab)^(0.5)

A.T.Q,

(2ab/(a+b))÷((ab)^(0.5)) = 12/13

or, 4(a^2)(b^2)/(ab(a+b)^2) = 144/169

or, 4ab/(a+b)^2 = 144/169

or, (a+b)^2/ab = 169/36

or,36( a^2 + b^2 + 2ab) = 169ab

or, 36((a/b) + (b/a) + 2) = 169

or, 36 ((a/b) + (b/a)) = 97

or, a/b + b/a = 97/36

Let a/b = x,

Thus,

x + 1/x = 97/36

or, (x^2 + 1)/x = 97/36

or,36( x^2 + 1) = 97x

or, 36(x^2) - 97x + 36 = 0

or, x

= (-97 + ((97^2) - 4(36)×(36))^(0.5))/72 (Quadratic Formula)(I am taking + sign)

= 4/9(I think the nos. are reversed)

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