Question

The harmonic mean of two numbers is $4$. Thus arithmetical mean $'A'$ and geometrical mean satisfy the relation $2{A^2} + {G^2} = 27$. Find the sum of those numbers.

Answer 1

Hint:The harmonic mean is type of numerical average. It is calculated by dividing the number of observations by the reciprocal each number in the series.Thus the harmonic mean is the reciprocal of the arithmetic mean of the reciprocal.In such type of questions it is important for the students to understand the meaning terms given.In this question students need to classify the data as the events mentioned and total outcomes.The tricky part in this is to establish the relationship between the events and the outcomes.
FORMULA USED:
It two number be $x\,andy$, The arithmetical mean,
$
  A = \dfrac{{x + y}}{2} \\
  Geometrical\,\,mean,\,\,G = \sqrt {xy} \\
  Harmonic\,\,mean,\,\,\,\,H = \dfrac{{2xy}}{{x + y}} \\
$

Complete step-by-step solution :
Let us understand the question here,
The question provides us with the harmonic mean of two numbers which is $4$.
 Thus arithmetical mean $'A'$ and geometrical mean satisfy the relation $2{A^2} + {G^2} = 27$.
The questions demand us to find the sum of those numbers.
Let the two numbers be $x\,andy$.
Let us recall the formula mentioned above ,
The arithmetical mean,
$
\Rightarrow A = \dfrac{{x + y}}{2} \\
\Rightarrow Geometrical\,\,mean,\,\,G = \sqrt {xy} \\
  \Rightarrow Harmonic\,\,mean,\,\,\,\,H = \dfrac{{2xy}}{{x + y}} \\
$
Now, substitute the values that we have got, we get
\[
\Rightarrow H = 4 \\
 \Rightarrow \dfrac{{{2}xy}}{{x + y}} ={4} \\
\Rightarrow \dfrac{{xy}}{{x + y}} = 2 \\
\Rightarrow xy = 2x + 2y\,\,\,\,\,\,\,\,\,\,\,\, - (i) \\
    \\
  Now\,it\,is\,given \\
  \Rightarrow 2A + {G^2} = 27 \\
\Rightarrow 2\dfrac{{(x + y)}}{2} + {(\sqrt {xy} )^2} = 27 \\
  \Rightarrow x + y + xy = 27 \\
\]
Further solving the equation we get,
Now we will use this information in $(ii)$
$
\Rightarrow x + y + 2x + xy = 27 \\
\Rightarrow 3x + 3y = 27 \\
\Rightarrow x + y = 9\,\,\,\,\,\,\,\,\,\,\,\,\, - (iii) \\
$
Now we will put this information in (i)
\[
\Rightarrow xy = 2x + 2y = 2(x + y) = 2 \times 9 = 18 \\
\Rightarrow xy = 18\,\,\,\,\,\,\,\,\, - (iv) \\
    \\
  now \\
\Rightarrow x - y = \sqrt {{{(x + y)}^2} - 4xy} \\
\Rightarrow x - y = \sqrt {{g^2} - 4 \times 18} \\
\Rightarrow x - y = \sqrt {81 - 72} \\
\Rightarrow x - y = \sqrt 9 \\
\Rightarrow x - y = 3 - (v) \\
    \\
On\, adding \,(iii) \,(v) \\
  \,\,\,\,\,\,\,\,x + y = 9 \\
  \,\,\,\,\,\,\,\,\underline {x - y = 3} \\
    \,\,\,\,\,\,\,\,2x = 12 \\
\Rightarrow \,\,\,\,\,\,\,x = \dfrac{{12}}{2} \\
\Rightarrow \,\,\,\,\,\,\,x = 6 \\
  put\,in\,(i) \\
\Rightarrow x + y = 9 \\
 \Rightarrow 6 + y = 9 \\
 \Rightarrow y = 3 \\
 \]
\[
  The\,\,required\,\,numbers\,\,are\, \\
\Rightarrow x = 6 \\
\Rightarrow y = 3 \\
\]
As the question demands the sum of those numbers,
We add x and y.
$x + y = 6 + 3 = 9$
Thus, 9 is the sum of the numbers.

Note:While attempting this question one just needs to remember the formulas used in the calculation. The student should not confuse harmonic mean, Arithmetic mean, and Geometric mean.
3.Students are advised to solve equations step by step and assign them numbers to avoid any type of calculation mistake.