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A set of five integers includes 30,45,75,75 and 100. When a sixth integer is added, the mean of the integers does not change. Which of the following is the sixth integer?
  & A.45 \\
 & B.50 \\
 & C.65 \\
 & D.75 \\

Last updated date: 20th May 2024
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Hint: In order to find the sixth integer in such a way that the mean does not change, firstly we must find out the mean of given five integers. After obtaining the mean, we must add a variable value as a number to the five integers and equate it to the obtained mean and find the value of the variable number. This would be our required answer.

Complete step-by-step solution:
Now let us learn about linear equations. A linear equation can be expressed in the form of any number of variables as required. As the number of the variables increase, the name of the equation simply denotes it. The general equation of a linear equation in a single variable is \[ax+b=0\]. We can find the linear equation in three major ways. They are: point-slope form, standard form and slope-intercept form.
Now let us find the sixth integer.
Firstly, let us find the mean of given five integers i.e. \[30,45,75,75\] and \[100\]
  & ~\Rightarrow \overset{-}{\mathop{x}}\,=\dfrac{30+45+75+75+100}{5} \\
 & ~\Rightarrow \overset{-}{\mathop{x}}\,=\dfrac{325}{5}=65 \\
\[\therefore \] The mean of given integers is \[65\].
Now let us consider the sixth integer as \[x\].
The mean of six integers including \[x\] would be,
  & ~\Rightarrow 65=\dfrac{30+45+75+75+100+x}{6} \\
 & ~\Rightarrow 65\times 6=325+x \\
 & ~\Rightarrow 390=325+x \\
 & ~\Rightarrow 390-325=x \\
 & ~\Rightarrow x=65 \\
\[\therefore \] The sixth integer such that the mean of given five integers does not change is \[65\].

Note: We have considered \[\overset{-}{\mathop{x}}\,\] as \[65\] in the second case as we have obtained the mean of five integers as \[65\]. We must have a note that the unknown integer can be placed anywhere in the list while finding the integer.

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