Question

The mean of six numbers is 20 If one number is excluded the mean is 15 Then the value of the excluded number is
A) 40
B) 42
C) 45
D) 48

Answer 1

Hint:
Here we will use the basic formula of the mean. First, we will use the formula of the mean and find the value of the sum of the six numbers. Then again we will use the formula of the mean and find the value of the sum of the five numbers. Then by subtracting the value of the sum of the five numbers from the sum of the six numbers, we will get the value of the excluded number.
Formula used:
We will use the formula Mean \[ = \] Sum of all numbers \[ \div \] Total number of observations.

Complete Step by Step Solution:
It is given that the mean of the six numbers is 20.
Now we will use the formula of the mean and we will get the sum of the six numbers. Therefore, we get
Mean \[ = \] Sum of 6 numbers \[ \div 6 = 20\]
Now by solving the above equation we will get the sum of the six numbers. Therefore, we get
Sum of six numbers \[ = 20 \times 6 = 120\]…………………… \[\left( 1 \right)\]
It is also given that if one number is excluded the mean is 15 which means the average of the five numbers out of those six numbers is equal to 15. Therefore, we get
Mean \[ = \] Sum of 5 numbers \[ \div 5 = 15\]
Now by solving the above equation we will get the sum of the five numbers. Therefore, we get
Sum of five numbers \[ = 15 \times 5 = 75\]……………………\[\left( 2 \right)\]
Now by subtracting the equation \[\left( 2 \right)\] from equation \[\left( 1 \right)\] we will get the value of the excluded number. Therefore, we get
Excluded number \[ = 120 - 75 = 45\]
Hence the value of the excluded number is 45.

So, option C is the correct option.

Note:
Mean is equal to the ratio of sum of the total numbers and total count of the numbers. Mean is also known as the average of the numbers. We should not confuse the mean with the median. Median is the middle value of the given list of numbers or it is the value which is separating the data into two halves i.e. upper half and lower half. We should also remember the formula of the mean when the data with the frequencies is given.
Mean \[ = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}\] where, \[{f_i}\] is the frequency of the class and \[{x_i}\] is the mid-value of the class interval.