Question

The mean of $5$ numbers is $18$ . If one number is excluded, their mean is $16$ . Find the excluded number.

Answer 1

Hint:
Mean of a set of numbers can be defined as the division of the sum of numbers by the number of quantities. Use this relation to find the sum of the initial five numbers. Now assume a variable ‘m’ for the number which is excluded from these five. The sum of the rest four numbers can be found by subtracting the number ‘m’ from the sum of all five numbers. Now use the definition of mean again to find a relation with one unknown, i.e. ’m’. Solve it to find the required answer.

Complete step by step solution:
Here in this problem, it is given that the mean of a group of five numbers is $18$. Then one of the numbers is removed and then the new mean becomes $16$ . With this information, we need to find the number which was removed.
Before starting with the solution, we should first understand the concept of mean that is used here. Mean, in mathematics, is the average of all the number present in the set or calculated “central” value of a set of numbers. The mean is calculated by dividing the sum of the numbers by the total number of quantities.
$ \Rightarrow {\text{Mean = }}\dfrac{{{\text{Sum of all the observations}}}}{{{\text{Total number of observations}}}}$
According to the given information in the question, using the above formula, we have:
$ \Rightarrow {\text{Mean = }}\dfrac{{{\text{Sum of all the observations}}}}{{{\text{Total number of observations}}}} \Rightarrow 18 = \dfrac{{{\text{Sum of five numbers}}}}{5}$
After transposing the unknown to one side, we can find the sum of these five numbers:
$ \Rightarrow 18 = \dfrac{{{\text{Sum of five numbers}}}}{5} \Rightarrow {\text{Sum of five numbers}} = 18 \times 5 = 90$
So we have now the sum of these five numbers as $90$ .
Let us assume that the number that is excluded from these five numbers be $'m'$
Therefore, the sum of the remaining four numbers will be equal to the number $'m'$ subtracted from the sum of five numbers.
$ \Rightarrow $ Sum of the remaining four numbers $ = $ Sum of all five numbers $ - $ Excluded number
After substituting the values in it, we get:
$ \Rightarrow $ Sum of four numbers\[ = \left( {90 - m} \right)\]
Now, let’s use the mean formula for new conditions, and we will find:
$ \Rightarrow {\text{Mean = }}\dfrac{{{\text{Sum of four numbers}}}}{{{\text{Total number of quantities}}}} \Rightarrow 16 = \dfrac{{90 - m}}{4}$
We can solve the above equation for finding the values for only unknown:

$ \Rightarrow 16 = \dfrac{{90 - m}}{4} \Rightarrow 90 - m = 16 \times 4 \Rightarrow m = 90 - 64 = 26$

Note:
In questions like this, using the definition of mean plays a crucial role in the solution. Be careful with the substitution of values in the formula ${\text{,i}}{\text{.e}}{\text{. Mean = }}\dfrac{{{\text{Sum of all the observations}}}}{{{\text{Total number of observations}}}}$
An alternative approach for the same problem is to find the sum of five numbers and the sum of four numbers separately using the mean formula. The difference between these two sums will be the excluded number.