Question

If sum of A.M and H.M between two positive number is 25 and their G.M is 12 then sum of number is
A. 9
B. 18
C. 32
D. 18 or 32

Answer 1

Hint:Take two variables for Arithmetic mean as well Harmonic Mean and find two equations using A.M and H.M formula. Try to find another one with a geometric mean’s formula which is given by the product of two numbers. compare both to find the function obtained to get the value of sum rather than finding individual value of the 2 numbers.

Complete step-by-step answer:
 Let’s begin with what information is given first one is that the arithmetic mean and harmonic mean sum is having a value 25 secondly, we have geometric mean is given as 12.
If we assume, those two no’s as a and b we get those A.M as
\[\dfrac{{a + b}}{2} = {\text{A}}{\text{.M}}\]
And \[{\text{H}}{\text{.M}} = \dfrac{{2ab}}{{a + b}},{\text{ G}}{\text{.M}} = \sqrt {ab} \]
So, we have \[{\text{GM}} = 12 = \sqrt {ab} \]
then \[ab = 144\] by squaring both side
other side we hade
\[{\text{A}}{\text{.M}} + {\text{H}}{\text{.M}} = 25\]
Squaring both the side we get
\[{\left( {{\text{A}}{\text{.M}} + {\text{ H}}{\text{.M}}} \right)^2} = {\text{625}}\]
We know that \[{\left( {a + b} \right)^2}\] can be return as \[{\left( {a b} \right)^2}{\text{ }} + {\text{ 4ab}}\] similarly we can write
\[{\left( {{\text{A}}{\text{.M}}-{\text{H}}{\text{.M}}} \right)^2} + 4{\text{ A}}{\text{.M}} \times {\text{H}}{\text{.M}} = 625\]
\[{\left( {{\text{A}}{\text{.M}}-{\text{H}}{\text{.M}}} \right)^2} = 625 - 4{\text{ A}}{\text{.M}} \times {\text{H}}{\text{.M }}\]
As we can see, we need \[{\text{A}}{\text{.M}} \times {\text{H}}{\text{.M }}\]
\[ = \dfrac{{a + b}}{2} \times \dfrac{{2ab}}{{a + b}} = ab = {\text{GM*2}}\]
\[\begin{gathered}
  \therefore {\text{ }}{\left( {{\text{A}}{\text{.M}} - {\text{H}}{\text{.M}}} \right)^2} = 625{\text{ }} - {\text{ }}4 \times 144 \\
   = {\text{ }}49 \\
\end{gathered} \]
So, \[{\text{A}}{\text{.M}} - {\text{H}}{\text{.M}} = 7\]
And we knew \[{\text{A}}{\text{.M}} + {\text{H}}{\text{.M}} = {\text{25}}\]
So, by adding 2 we get
\[\dfrac{{\begin{array}{*{20}{c}}
  {{\text{A}}{\text{.M}} + {\text{H}}{\text{.M}}} \\
  {{\text{A}}{\text{.M}} - {\text{H}}{\text{.M}}}
\end{array}}}{{2{\text{ A}}{\text{.M}}}}\begin{array}{*{20}{c}}
  {\begin{array}{*{20}{c}}
  {\begin{array}{*{20}{c}}
  { = {\text{ }}25} \\
  { = {\text{ 7}}}
\end{array}} \\
  { = {\text{ }}32}
\end{array}} \\
  {}
\end{array}\] by dividing both side with 2 we get \[{\text{A}}{\text{.M}} = 16\]
as we know that A.M is Sum by 2 of the two values. i.e., \[\dfrac{{a + b}}{2}\]
so \[a + b = 2 \times {\text{A}}{\text{.M}} = 32\]

Hence, option C is the correct answer.

Note: Solving variable and finding values has no use. as we need a sum of two values and the Arithmetic mean is given by the average of the number. Which means half of the sum of the numbers. We can try on A.M direct value and use relations with Arithmetic Mean.