Question

If the mean of $3,7,11$ and $x$ is $9$. What is the value of $x$? divide \[\left( {10{x^3} + 24{x^2}} \right).6x\]

Answer 1

Hint: We are given with the observations and the mean. So we will use the formula to find the missed observation or the x term. Then we will put this value of x in the equation so given and then we will calculate the total value of the expression.

Formula used:
\[mean = \dfrac{{Sum{\text{ }}of{\text{ }}observations}}{{Total{\text{ }}number{\text{ }}of{\text{ }}observations}}\]

Complete step by step solution:
Given the observations are 3,7, 11 and 9.
Also given the mean is 9. So we will directly use the formula of mean.
\[mean = \dfrac{{Sum{\text{ }}of{\text{ }}observations}}{{Total{\text{ }}number{\text{ }}of{\text{ }}observations}}\]
Now putting the respective values that are observations in numerator and total number in denominator. Provided the mean is also given,
\[9 = \dfrac{{3 + 7 + 11 + x}}{4}\]
On cross multiplying,
\[9 \times 4 = 3 + 7 + 11 + x\]
Taking the product and the sum on respective sides,
\[36 = 21 + x\]
On taking the constants on one side we get,
\[36 - 21 = x\]
Calculating the difference,
\[x = 15\]
Thus this is the value of x.
Now the value of expression will be,
\[\left( {10{x^3} + 24{x^2}} \right).6x\]
Putting the value of x as 15,
\[ = \left( {10{{\left( {15} \right)}^3} + 24{{\left( {15} \right)}^2}} \right).6\left( {15} \right)\]
Taking the cube and square,
\[ = \left( {10 \times 3375 + 24 \times 225} \right) \times 80\]
On multiplying we get,
\[ = \left( {33750 + 5400} \right) \times 80\]
On adding the numbers in the bracket,
\[ = 39150 \times 80\]
On multiplying we get,
\[ = 3132000\]
This is the answer to the expression.
Therefore, The value of $x=15$ and the value of \[\left( {10{x^3} + 24{x^2}} \right).6x=3132000\].

Note:
To find the mean we have a simple formula. Then that value of x is used in the expression. There are various other methods to find the mean. Mean, median and mode are the terms related to statistics. In data integrations these terms are used.