The mean of 21 test score is 78. What mean score must three 
additional students acheive so that the mean for all 24 
students is 80. 

Let x = the mean score of the three additional students

We use this formula three times:

 = , or 

=

First we use it to find the sum of the scores when there 
are 21 students, by substituting M=78 and N=21 and 
solving for S:

=

 = 

Mutiply both sides by 21

 = 

So, the sum of scores of the 21 is 1638

Second we use it to find the sum of the scores of the three
additional students, by substituting M=x and N=3 and 
solving for S:

=

 = 

Mutiply both sides by 3

 = 

So the sum of the scores of the 3 additional students is 3x.

Thirdly we use it to find the sum of the scores of all 24
additional students, by substituting M=80 and N=24 and 
solving for S:

=

 = 

Mutiply both sides by 24

 = 
 
S the sum of the scores of all 24 students 

Now we form our equation from:

(Sum of scores of the 21) + (Sum of scores of the 3) = (Sum of scores of the 24)   

                                           1638 + 3x = 1920

Solve that and get x = 94.  So the three additional students must
average 94.

Edwin