Question

Find the difference between two natural number of their GM is 18 and HM is 14.4

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{Geometric mean of two natural no. is 18 and}$

$\textsf{their Harmonic mean is 14.4}$

$\underline{\textbf{To find:}}$

$\textsf{The difference between the two natural no.}$

$\underline{\textbf{Solution:}}$

$\textsf{Let a and b be the given  two natural number with a > b}$

$\mathsf{Geometric\;mean=18}$

$\implies\mathsf{\sqrt{ab}=18}$

$\implies\mathsf{ab=18^2}$

$\implies\mathsf{ab=324}$

$\mathsf{Harmonic\;mean=14.4}$

$\implies\mathsf{\dfrac{2\,ab}{a+b}=14.4}$

$\implies\mathsf{\dfrac{ab}{a+b}=7.2}$

$\implies\mathsf{\dfrac{324}{a+b}=7.2}$

$\implies\mathsf{a+b=\dfrac{324}{7.2}}$

$\implies\mathsf{a+b=45}$

$\textsf{We know that,}$

$\boxed{\mathsf{(a-b)^2=(a+b)^2-4ab}}$

$\mathsf{(a-b)^2=45^2-4(324)}$

$\mathsf{(a-b)^2=2025-1296}$

$\mathsf{(a-b)^2=729}$

$\mathsf{a-b=\sqrt{729}}$

$\implies\boxed{\mathsf{a-b=27}}$

$\therefore\textbf{The difference between the natural numbers is 27}$

$\underline{\textbf{Find more:}}$

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