Question

The arithmetic mean of 7, x-2, 10, x+3 is 9 Find x.

Answer 1

Hint – In this question use the direct formula for arithmetic mean which is $A.M = \dfrac{{a + b + c}}{3}$, if we are given three numbers, this same concept can be extended to 4 numbers as well.

Complete step-by-step answer:

Let us suppose three numbers a, b and c the arithmetic mean (A.M) of the numbers given as
$ \Rightarrow A.M = \dfrac{{a + b + c}}{3}$

Now it is given that the A.M of 7, (x – 2), 10 and (x + 3) is 9 so according to above property it is written as
$ \Rightarrow A.M = \dfrac{{7 + \left( {x - 2} \right) + 10 + \left( {x + 3} \right)}}{4} = 9$

Now simplify the above equation we have,
$ \Rightarrow 7 + \left( {x - 2} \right) + 10 + \left( {x + 3} \right) = 9 \times 4$
$ \Rightarrow 2x + 18 = 36$
$ \Rightarrow 2x = 36 - 18 = 18$

Now divide by 2 we have,
$ \Rightarrow x = \dfrac{{18}}{2} = 9$

So the required value of x is 9.

So this is the required answer.

Note – There are in general two very frequently used means, the first one is arithmetic mean and the other is geometric means. Arithmetic mean is the average of a set of numerical value, calculated by adding them together and dividing them by the total number, whereas geometrical mean indicates the central tendency of a set of numbers, in other words it is defined as the nth root of the product of n terms, that is for ${x_1},{x_2},{x_3}...................{x_n}$ geometric mean is defined as $^n\sqrt {{x_1},{x_2},{x_3}...................{x_n}} $.