The harmonic mean between two numbers is 14fr | vedclass

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Question

(A) Let the two numbers be $a$ and $b$.We know that the relationship between Arithmetic Mean $(A)$,Geometric Mean $(G)$,and Harmonic Mean $(H)$ is $G^

Vedclass · Accepted Answer

$72$

Vedclass · Answer

(A) Let the two numbers be $a$ and $b$.We know that the relationship between Arithmetic Mean $(A)$,Geometric Mean $(G)$,and Harmonic Mean $(H)$ is $G^2 = AH$.Given $H = 14\frac{2}{5} = \frac{72}{5}$ and $G = 24$.Substituting the values: $(24)^2 = A 	imes \frac{72}{5}$.$576 = A 	imes \frac{72}{5} \Rightarrow A = \frac{576 	imes 5}{72} = 8 	imes 5 = 40$.Since $A = \frac{a+b}{2} = 40$,we have $a+b = 80$ $(i)$.Since $G = \sqrt{ab} = 24$,we have $ab = 576$ $(ii)$.Using the identity $(a-b)^2 = (a+b)^2 - 4ab$,we get $(a-b)^2 = (80)^2 - 4(576) = 6400 - 2304 = 4096$.Thus,$a-b = \sqrt{4096} = 64$ $(iii)$.Solving equations $(i)$ and $(iii)$ by adding them: $2a = 144 \Rightarrow a = 72$.Substituting $a=72$ into $(i)$: $72 + b = 80 \Rightarrow b = 8$.The greater number is $72$.

The harmonic mean between two numbers is $14\frac{2}{5}$ and the geometric mean is $24$. The greater number among them is

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