Question

The arithmetic mean of two numbers is 6 and their geometric mean G and harmonic mean H satisfy the relation $ {G^2} + 3H = 48 $ . Find the two numbers

Answer 1

Hint: In order to solve this question we will write the terms arithmetic mean in form of equation then we will write the whole equation which we have to prove in terms of their formula then we will put all the known values to this equation then we will get a relation between a and b. Since we have on relation in a and b this will be another and like this we have two equations and two variables so we will get a and b.

Complete step by step solution:
For solving this question we will first let the two numbers be a and b:
Since it is given to us that the arithmetic mean of these two numbers is 6 so:
 $ \dfrac{{a + b}}{2} = 6 $
So from here we will get the relation:
 $ a + b = 12 $ …………………(1)
And we know that:
 $ G = {\left( {ab} \right)^{\dfrac{1}{2}}} $
And:
 $ H = \dfrac{{2ab}}{{a + b}} $
Now according to the question it is given that:
 $ {G^2} + 3H = 48 $
If we will write this in terms of the formula we will get:
 $ {\left( {{{\left( {ab} \right)}^{\dfrac{1}{2}}}} \right)^2} + 3\left( {\dfrac{{2ab}}{{a + b}}} \right) = 48 $
Now putting the value of a+b from equation 1 we will get:
 $ {\left( {{{\left( {ab} \right)}^{\dfrac{1}{2}}}} \right)^2} + 3\left( {\dfrac{{2ab}}{{12}}} \right) = 48 $
Now on further solving this we will get:
 $ ab + \dfrac{1}{2}ab = 48 $
On adding right side we will get:
 $ \dfrac{3}{2}ab = 48 $
From here we will get the value of:
 $ ab = 32 $
Now putting the value of a = 12-b we will get:
 $ \left( {12 - b} \right)b = 32 $
On further solving this we will get:
 $ {b^2} - 12b + 32 = 0 $
 $ (b-8) (b-4) =0 $
By solving this quadratic equation we will get the value of b = 8 so the value of a will be 4 OR if b=4 then a=8
So, the correct answer is “a=4 AND b=8 OR a=8 AND b=4”.

Note: While solving this question we let me tell you one thing that at the end when we solve the quadratic equation we will get two values we are taking only one it is because the other value will be 16 to maintain we will get the value of a negative. And we know that the product of two numbers if one is positive and the other one is negative will always be negative, so the 16 is not possible in this question.