Question

The average test scores of a group of students was 72marks.Three students were left from the group whose average marks was85.The average matks of yhe remaining students became 69marks.Find out how many students were there in the group at first?

Answer 1

There were 16 students in the group at first.

Step-by-step explanation:

We are given that the average test scores of a group of students was 72 marks. Three students were left from the group whose average marks was 85. The average marks of the remaining students became 69 marks.

Let the number of students in the group at first be n.

AS we know that the formula for finding the average for group of numbers is given as;

        Average (Mean) = $\frac{\sum X}{n}$

where, $\sum X$ = Sum of all values in the data

               n = Total number of observations in the data

• So, Firstly we are given that the average test scores of a group of students (total) was 72 marks, i.e;

         Average = $\frac{\sum X_n}{n}$

                 72   =  $\frac{\sum X_n}{n}$  

           $\sum X_n = 72 \times n$    ---------- {equation 1}

Here $\sum X_n$ represents the sum of test scores of all students in the group.                                      

• Now, three students were left from the group whose average marks was 85, i.e;

        Average = $\frac{\sum X_3}{3}$

                 85  =  $\frac{\sum X_3}{3}$  

           $\sum X_3 = 85 \times 3$      

           $\sum X_3 = 255$        ---------- {equation 2}  

Here $\sum X_3$ represents the sum of test scores of three students in the group.

• Finally, the average marks of the remaining students became 69 marks, i.e.;

        Average = $\frac{\sum X_n_-_3}{n-3}$

                 69  =  $\frac{\sum X_n - \sum X_3}{n-3}$  

                 69  = $\frac{72n - 255}{n-3}$       {using equation and 2}                  

Here $\sum X_n_-_3$ represents the sum of test scores of remaining students in the group.

So,            69  = $\frac{72n - 255}{n-3}$

                 $69(n-3) = 72n-255$

                 $69n - 207=72n-255$

                 $72n-69n=255-207$

                      $3n=48$

                       $n = \frac{48}{3} = 16$

Therefore, there were 16 students in the group at first.