Question

To increase the mean of 4 numbers by 2, by how much would the sum of the 4 numbers have to increase ?

Answer 1

Hint: Firstly, we must know what actually the term means. Mean is found by adding all the numbers in the data set and dividing it by the total number of numbers present in the data set. The mean $\bar x$ is found using the formula $\bar x = \dfrac{1}{n}\sum\limits_{i = 1}^n {{x_i}} $. We make use of this formula to the given data and add 2 on both sides. Then we simplify it to obtain the desired answer.

Complete step by step solution:
Let us firstly understand the definition of mean of numbers.
The mean (arithmetic mean) is found by adding all the values given and dividing it by the total number of values present.
The mean is commonly called the average.
If ${x_1},{x_2},....,{x_n}$ are the values of variable x, then the mean is usually denoted by $\bar x$ and is given by,
$\bar x = \dfrac{{{x_1} + {x_2} + {x_3} + .... + {x_n}}}{n}$
$ \Rightarrow \bar x = \dfrac{1}{n}\sum\limits_{i = 1}^n {{x_i}} $ …… (1)
Where, $\sum {{x_i}} $ represents the sum of all numbers
               $n$ represents the number of values
                ${x_i}$ represents the given values
In the above question, we have given a set of 4 numbers and the mean of these is found using the formula given in the equation (1).
The mean of 4 numbers can be denoted by,
 $\bar x = \dfrac{1}{4}\sum\limits_{i = 1}^4 {{x_i}} $
Since the mean is increased by 2, we add the number 2 to both sides we get,
$ \Rightarrow \bar x + 2 = \dfrac{1}{4}\sum\limits_{i = 1}^4 {{x_i}} + 2$
This can also be written as,
$ \Rightarrow \bar x + 2 = \dfrac{{\sum\limits_{i = 1}^4 {{x_i}} }}{4} + 2$
Taking LCM on the right hand side we get,
$ \Rightarrow \bar x + 2 = \dfrac{{\sum\limits_{i = 1}^4 {{x_i}} + 8}}{4}$
Now multiplying by 4 on both sides we get,
$ \Rightarrow 4 \times (\bar x + 2) = \dfrac{{\sum\limits_{i = 1}^4 {{x_i}} + 8}}{4} \times 4$
$ \Rightarrow 4(\bar x + 2) = \sum\limits_{i = 1}^4 {{x_i}} + 8$
Hence we note that to get an increment of 2 in the mean of 4 numbers, the sum of the four numbers would have to increase by 8.

Thus, the answer for the given question is 8.

Note :
Students must be familiar with the concept of mean, median and mode of the given values.
Sometimes the problem can be given in terms of the median or mode also.
Students must know the formula to find out the mean and understand the definition of it correctly. The mean also known as average is found by taking the sum of the values and dividing it by the count of the values.
The mean $\bar x$ is found using the formula $\bar x = \dfrac{1}{n}\sum\limits_{i = 1}^n {{x_i}} $
Where, $\sum {{x_i}} $ represents the sum of all numbers
               $n$ represents the number of values
                ${x_i}$ represents the given values
Now we learn what the definition of median says.
If the items are arranged in ascending or descending order of magnitude, then the middle value is called Median.
The mode is that value in a series of observations which occurs with greatest frequency.