Question

Let $ x,y $ be two positive numbers such that their Arithmetic and Harmonic means are $ 9,36 $ respectively. Choose the correct option for the geometric mean of given numbers.
1) $ 18 $
2) $ 12 $
3) $ 16 $
4) None of the above.

Answer 1

Hint: We have given the Arithmetic and Harmonic means to find the geometric mean. We can solve this problem in three ways. One is finding the given two numbers from two means or manipulating the given two means or using a direct formula. Now we are going to do the manipulating method in this way we can get the answer quickly without remembering the relation between means.

Complete step-by-step answer:
Given that,
 $ x,y $ are two positive numbers, and
Arithmetic mean of $ x,y $ , $ $ A = 9 $ $ ,
Harmonic mean of $ x,y $ , $ H = 36 $ ,
Let the geometric mean of $ x,y $ be $ G $
We know that,
Arithmetic mean of $ x,y $ $ = \dfrac{{x + y}}{2} $ ,
Harmonic mean of $ x,y $ $ = \dfrac{{2xy}}{{x + y}} $ .
Therefore,
 $ \Rightarrow \dfrac{{x + y}}{2} = 9,\dfrac{{2xy}}{{x + y}} = 36 $ ,
On some calculations, we get
 $ \Rightarrow x + y = 18,\dfrac{{2xy}}{{x + y}} = 36 $
Now combine both equations, we get
 $ \Rightarrow \dfrac{{2xy}}{{18}} = 36 $
After calculations, we get
 $ xy = 384 $ .
Now we have both $ x + y,xy $ so we can calculate $ x,y $ but we can have G.M without calculating them.
We know that,
Geometric Mean of $ x,y $ , $ G = \sqrt {xy} $
Now we know the value of $ xy $ letting us substitute it in $ G $ to get the answer.
 $ \Rightarrow G = \sqrt {384} = \sqrt {{{18}^2}} $
 $ \Rightarrow G = 18 $
So the required answer is $ 18 $ .
Therefore the correct option is 1.
So, the correct answer is “Option 1”.

Note: This problem is very easy when we remember the formula. But we should be able to solve it without using it in time. Time is also a key factor during exams. If we remember the formula there is less chance of making an error but in this method, there is a little chance of making an error. We should always remember the formulas of all means because those are the definitions of them and we cannot derive them.