If the sum of the arithmetic mean and the har | vedclass

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(C) Let the two positive numbers be $a$ and $b$.Arithmetic Mean $A = \frac{a+b}{2}$,Geometric Mean $G = \sqrt{ab} = 12$,Harmonic Mean $H = \frac{2ab}{

Vedclass · Accepted Answer

$32$

Vedclass · Answer

(C) Let the two positive numbers be $a$ and $b$.Arithmetic Mean $A = \frac{a+b}{2}$,Geometric Mean $G = \sqrt{ab} = 12$,Harmonic Mean $H = \frac{2ab}{a+b}$.Given $A + H = 25$ and $G = 12$.Since $G^2 = AH$,we have $AH = 12^2 = 144$.Substituting $H = \frac{144}{A}$ into $A + H = 25$:$A + \frac{144}{A} = 25 \implies A^2 - 25A + 144 = 0$.$(A - 16)(A - 9) = 0$,so $A = 16$ or $A = 9$.Since $A \geq G$,we must have $A = 16$.Thus,$\frac{a+b}{2} = 16 \implies a+b = 32$.

If the sum of the arithmetic mean and the harmonic mean between two positive numbers is $25$ and their geometric mean is $12$,then what is the sum of the numbers?

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