Question

If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers

Answer 1

Hint: we need to know the general formula of Arithmetic and Geometric means of two numbers.
It is given that a and b are two numbers.
Arithmetic mean (AM) of the given two numbers is $\frac{{a + b}}{2} = 10$ ... (1)
Geometric Mean (GM) of the given two numbers is $\sqrt {ab} = 8$ ... (2)
On simplifying equation (1) and (2), we get
$ \Rightarrow a + b = 20$ and $ab = 64$
We need to find the unique values of a and b.
Clearly, a and b are roots of the quadratic equation
${x^2} - (a + b)x + ab = 0$
Substituting $\left( {a + b} \right)\&$ ab values
${x^2} - 20x + 64 = 0$
Factorization of the above quadratic equation to find roots
$ \Rightarrow (x - 16)(x - 4) = 0$
$ \Rightarrow x = 4,16$
$\therefore a = 4,b = 16$ or $a = 16,b = 4$ are the required values.

Note: Arithmetic mean is the sum of a collection of all numbers divided by count of numbers in the collection. Geometric mean indicates the central tendency or typical value of a set of numbers by using the product of their values.