The Geometric Mean (G.M.) and Harmonic Mean ( | vedclass

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Question

(A) Let the two numbers be $a$ and $b$. Given that the $G.M. = \sqrt{ab} = 10$, so $ab = 100$. Given that the $H.M. = \frac{2ab}{a+b} = 8$. Substituti

Vedclass · Accepted Answer

$5, 20$

Vedclass · Answer

(A) Let the two numbers be $a$ and $b$. Given that the $G.M. = \sqrt{ab} = 10$, so $ab = 100$. Given that the $H.M. = \frac{2ab}{a+b} = 8$. Substituting $ab = 100$ into the $H.M.$ equation: $\frac{2(100)}{a+b} = 8$ $\Rightarrow \frac{200}{a+b} = 8$ $\Rightarrow a+b = \frac{200}{8} = 25$. Now we have $a+b = 25$ and $ab = 100$. The quadratic equation with roots $a$ and $b$ is $x^2 - (a+b)x + ab = 0$. $x^2 - 25x + 100 = 0$. $(x-20)(x-5) = 0$. Thus, the numbers are $5$ and $20$.

The Geometric Mean $(G.M.)$ and Harmonic Mean $(H.M.)$ of two numbers are $10$ and $8$ respectively. The numbers are:

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