Question

The mean of five numbers is \[27\] . If one of the numbers is excluded the mean gets reduced by \[2\] . The excluded number is
\[\left( 1 \right)\] \[35\]
\[\left( 2 \right)\] \[27\]
\[\left( 3 \right)\] \[25\]
\[\left( 4 \right)\] \[40\]

Answer 1

Hint: We have to find the value of the excluded number using the given condition for the value of the mean of five numbers. We will solve this question using the concept of mean of numbers. First we will write an expression for the given initial condition of mean for the five numbers and then we will make another expression using the given condition by taking the value of mean for the four values and taking the value of new mean as the difference of the initial mean value. Thus using the two equations, we can solve the equation using the concept of solving the linear equations and hence we will get the value of the excluded number.

Complete step-by-step solution:
Given :
The mean of five numbers is \[27\]. If one of the numbers is excluded the mean gets
reduced by \[2\].
Let us consider that the mean of five numbers is given as :
\[Mean = \dfrac{{{x_1} + {x_2} + {x_3} + {x_4} + {x_5}}}{5}\]
Hence , the expression of mean value is written as:
\[\dfrac{{{x_1} + {x_2} + {x_3} + {x_4} + {x_5}}}{5} = 27\]
So , we get the sum of the five numbers as :
\[{x_1} + {x_2} + {x_3} + {x_4} + {x_5} = 27 \times 5\]
\[{x_1} + {x_2} + {x_3} + {x_4} + {x_5} = 135 - - - - (1)\]
Now , it is given that the value of mean is reduced by \[2\] when a number is excluded from the given five numbers , the mean can be written as :
\[New{\text{ }}mean = 27 - 2\]
\[New{\text{ }}mean = 25\]
The expression for new mean can be written as :
\[New{\text{ }}mean = \dfrac{{{x_1} + {x_2} + {x_3} + {x_4}}}{4}\]
\[\dfrac{{{x_1} + {x_2} + {x_3} + {x_4}}}{4} = 25\]
The value of sum of new mean can be written as :
\[{x_1} + {x_2} + {x_3} + {x_4} = 25 \times 4\]
\[{x_1} + {x_2} + {x_3} + {x_4} = 100 - - - - (2)\]
Now , to get the value of the excluded number we can solve the value by subtracting equation \[\left( 2 \right)\] from equation \[\left( 1 \right)\] as:
\[{x_1} + {x_2} + {x_3} + {x_4} + {x_5} - \left( {{x_1} + {x_2} + {x_3} + {x_4}} \right) = 135 - 100\]
On solving , we get
\[{x_5} = 35\]
Thus , the value of the excluded number is \[35\] .
Hence , the correct option is \[\left( 1 \right)\] .

Note: The formula of mean of n number of terms is given as : the sum of the value of the \[n\] observations upon the total number of observations or simply the number of terms . This formula can be only used for single number data , such as the marks of students in a class , the height of students in a college etc . We have different formulas for different values of the data.