Question

16. Arithmetic mean between two numbers is 5 and Geometric mean between

them is 4. Find the Harmonic mean between the numbers. 2

Answer 1

hope it helps you....

Answer 2

Solution:

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Given:

Statement 1 : Arithmetic mean of two numbers is 5,.

Let the two numbers be 'a & b',.

So,

We can say that,

=> $\frac{a+b}{2} = 5$ ...(i)

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Statement 2 :

Geometric mean of the two numbers is 4.

Which means,

=> $\sqrt{ab} = 4$

=> ab = 16 ..(ii)
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To find:

  The harmonic mean of the two numbers,

=> $\frac{2}{ \frac{1}{a} + \frac{1}{b} }$

=> $\frac{2}{( \frac{a+b}{ab}) }$

=> $2( \frac{ab}{a+b} )$

=> $\frac{2ab}{a+b}$...

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As we know that,

(i) => a + b = 10,

b = 10 - a...(iii),

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(ii) => ab = 16,.

=>a(10-a) = 16

=> 10a - a² = 16

=>  10a - a² - 16 = 0

=> -10a + a² + 16

=> a² - 10a + 16

=> (a - 8) (a - 2) = 0...(iv)
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As we know that,

 anything multiplied by 0 is 0,

So, for the equation (iv) to be true,

a - 8 = 0 (or) a - 2 = 0,.

So,

either  a = 8 (or) a = 2,.

If  a = 8,                l If a = 2,
                              l
=> b = 10 -a         l  => b = 10-a
                              l
= b = 10 - 8          l  => b = 10 - 2
                              l
=> b = 2        (or)    => b = 8,.

Hence,

Harmonic mean

=> $\frac{2ab}{a+b}$

Substituting value of a & b in the formula,

=> $\frac{2(8)(2)}{2 + 8}$

=> $\frac{32}{10}$ = 3.2

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                                      Hope it Helps!!