Question

the airthmetic mean of two observation is 127.5 and their geometric mean is 60. Find their harmonic mean and the two observation.

Answer 1

GIVEN :–

• The airthmetic mean of two observation is 127.5

• And their geometric mean is 60.

TO FIND :–

• Harmonic mean and the two observation = ?

SOLUTION :–

• Let's two observation are a and b.

• According to the first condition –

$\\ \bf \implies A.M. = \dfrac{a + b}{2}$

$\\ \bf \implies 127.5= \dfrac{a + b}{2}$

$\\ \bf \implies a + b = 2 \times 127.5$

$\\ \bf \implies a + b = 255$

$\\ \bf \implies b = 255 - a \: \: \: \: \: \: \: - - -eq.(1)$

• According to the second condition –

$\\ \bf \implies G.M. = \sqrt{ab}$

$\\ \bf \implies 60= \sqrt{ab}$

• Using eq.(1) –

$\\ \bf \implies 60= \sqrt{a(255 - a)}$

• Square on both side –

$\\ \bf \implies (60)^{2} = a(255 - a)$

$\\ \bf \implies 3600=255a - {a}^{2}$

$\\ \bf \implies {a}^{2} - 255a + 3600=0$

$\\ \bf \implies {a}^{2} - 240a - 15a + 3600=0$

$\\ \bf \implies a(a - 240) - 15(a - 240)=0$

$\\ \bf \implies (a - 15)(a - 240)=0$

$\\ \bf \implies a = 15,240$

• So that –

$\\ \bf \implies b = 240,15$

• We know that –

$\\ \bf \implies H.M. = \dfrac{2}{ \dfrac{1}{a} + \dfrac{1}{b} }$

$\\ \bf \implies H.M. = \dfrac{2}{ \dfrac{1}{15} + \dfrac{1}{240} }$

$\\ \bf \implies H.M. = \dfrac{2}{ \dfrac{240 + 15}{15 \times 240} }$

$\\ \bf \implies H.M. = \dfrac{2}{ \dfrac{255}{3600} }$

$\\ \bf \implies H.M. = \dfrac{7200}{255}$

$\\ \large \implies{ \boxed{ \bf H.M. =28.23}}$