Question

The average of $x$ numbers is $3x$. If $(x-1)$ is subtracted from each given number, what will be the new average?
(a) $3(x-1)$
(b) $2x+1$
(c) $2x-1$
(d) Data inadequate

Answer 1

Hint: Assume the numbers as ‘${{a}_{1}},{{a}_{2}},{{a}_{3}},...................{{a}_{x}}$’. Find their sum and divide by the total number of numbers that is ‘$x$’ to find the average. This will result in the formation of an equation. Now, subtract $(x-1)$ from each number and take their sum, then divide this sum by $x$ and use the relation obtained in the first step to simplify the expression and get the new average.


Complete step-by-step answer:
First, let us know the term ‘average’. For a data set, the arithmetic mean, also called the expected value or average, is the central value of a discrete set of numbers: specifically, the sum of values divided by the number of values. There are several kinds of meanings in mathematics. We will not take the discussion that far. We have to deal with arithmetic meanings.

Now, let us come to the question. We have been given the average of $x$. Let us denote the numbers as ${{a}_{1}},{{a}_{2}},{{a}_{3}},...................{{a}_{x}}$. Therefore,

$\begin{align}

  & mean=\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+...........+{{a}_{x}}}{x} \\

 & 3x=\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+...........+{{a}_{x}}}{x} \\

\end{align}$

By cross multiplication we get,

${{a}_{1}}+{{a}_{2}}+{{a}_{3}}+...........+{{a}_{x}}=3{{x}^{2}}..............(i)$

Now, subtracting $(x-1)$ from each term and taking their sum we get,

$\begin{align}

  & new\text{ }mean=\dfrac{\left( {{a}_{1}}-(x-1) \right)+\left( {{a}_{2}}-(x-1) \right)+\left( {{a}_{3}}-(x-1) \right)+........\left( {{a}_{x}}-(x-1) \right)}{x} \\

 & =\dfrac{({{a}_{1}}+{{a}_{2}}+{{a}_{3}}+.........{{a}_{x}})-(x+x+x+.....+x\text{ times)+(1+1+1+}......x\text{ times)}}{x} \\

 & \\

\end{align}$

Using equation (i) we can write,

$\begin{align}

  & new\text{ }mean=\dfrac{3{{x}^{2}}-{{x}^{2}}+x}{x} \\

 & =\dfrac{2{{x}^{2}}+x}{x} \\

 & =2x+1 \\

\end{align}$

Hence, option (b) is the correct answer.


Note: In the calculation of new mean, one can see that we have subtracted $(x-1)$ from each term and there are $x$ terms, so we can say that $x$is subtracted $x$ times and 1 is added $x$ times to the sum of initial numbers. That’s how we have proceeded for further calculation.