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 $\left(a+b\right)\mathrm{\&}$ 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.