Question

If the mean of x + 2, 2x + 3, 3x + 4 and 4x + 5 is x + 2 then find the value of x.
(a) 0
(b) 1
(c) -1
(d) 2

Answer 1

Hint: First take the sum of all the terms, then count the number of terms and assume it to be ‘n’. To find the mean apply the formula: - Mean = (Sum of terms)/ (Total number of terms). Substitute the value of mean given in the question and form a linear equation in one variable (x). Solve this equation to find the value of x.

Complete step-by-step answer:
We have been provided with the terms x + 2, 2x + 3, 3x + 4 and 4x + 5. The mean of these observations is given as x + 2. We have to find the value of ‘x’.
Now, we know that the mean or average of a list of numbers is the sum of all the numbers divided by the amount of numbers. It is also called Arithmetic mean. Mathematically,
\[\Rightarrow \overline{x}=\dfrac{1}{n}\left( \sum\limits_{i=1}^{n}{{{x}_{i}}} \right)=\dfrac{1}{n}\left( {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+.......+{{x}_{n}} \right)\]
In the above formula, ‘\[\overline{x}\]’ is the notation of mean, ‘n’ is the amount of numbers or observations, (\[{{x}_{1}},{{x}_{2}},.......,{{x}_{n}}\]) are the numbers or observations.
Now, let us come to the question.
On counting the given terms we find that there are four terms: - x + 2, 2x + 3, 3x + 4 and 4x + 5. So, the value of ‘n’ is 4.
Taking the sum of these terms, we get,
\[\Rightarrow \sum\limits_{i=1}^{4}{{{x}_{i}}}={{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}\]
\[\begin{align}
  & \Rightarrow \sum\limits_{i=1}^{4}{{{x}_{i}}}=\left( x+2 \right)+\left( 2x+3 \right)+\left( 3x+4 \right)+\left( 4x+5 \right) \\
 & \Rightarrow \sum\limits_{i=1}^{4}{{{x}_{i}}}=10x+14 \\
\end{align}\]
Now, applying the formula for mean, we get,
\[\Rightarrow \overline{x}=\dfrac{1}{n}\left( \sum\limits_{i=1}^{n}{{{x}_{i}}} \right)\]
Substituting the values of n and \[\sum\limits_{i=1}^{n}{{{x}_{i}}}\], we get,
\[\Rightarrow \overline{x}=\dfrac{1}{4}\times \left( 10x+4 \right)\]
Substituting the value of mean (\[\overline{x}\]) = x + 2, we get,
\[\Rightarrow x+2=\dfrac{10x+14}{4}\]
By cross – multiplication, we get,
\[\begin{align}
  & \Rightarrow 4x+8=10x+14 \\
 & \Rightarrow 10x-4x=8-14 \\
 & \Rightarrow 6x=-6 \\
\end{align}\]
Dividing both sides with 6, we get,
\[\Rightarrow x=-1\]

So, the correct answer is “Option c”.

Note: One must remember that, there are several kinds of means like arithmetic mean, geometric mean, harmonic mean and many more but when only the term ‘mean’ is given then we have to consider it as an arithmetic mean. It can also be called ‘average’. So, in the above question we have applied the formula of arithmetic mean, that is the ratio of sum of observations to the number of observations.