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

Answer

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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.

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.

Last updated date: 27th May 2023

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