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11)Ms. Forsythe gave the same algebra test to her three classes. The first class averaged 80%, the second class averaged 85%, and the third 89%. Together, the first two classes averaged 83%, and the second and third classes together averaged 87%. What was the average for all three classes combined? Express your answer to the nearest hundredth.

 

The average  =  total points / number of students

So....total points = average*number of students

 

Let the total number of students in  each class = x , y  and z

 

The total number of points the first class amassed  was   80x

The total number of points amased by the second class was  85y

The total number of points amassed by the third class = 89z

 

So...for the first two classes we have

 

[ 80x + 85y ] / [ x + y] = 83

80x + 85y = 83x + 83y  subtract  80x, 83y from both sides   

3x = 2y

x = 2/3y

 

And,,,for the second two classes we have

 

[ 85y + 89z ]  / [ y + z ]  =  87  

[  85y + 89z ]  = 87 [y + z]

85y + 89z = 87y + 87z      subtract  85y, 87z from both sides

2z = 2y 

y  = z

 

So...for the three classes.....

 

Total points by all three classes /  number of class members  = the average for all three classes

 

[ 80x + 85y  + 89z ] /  [ x + y + z ]   =

 

[sub for  x and z ] 

 

[80 (2/3)y  + 85y  + 89y]  /  [ (2/3)y  + y  + y ] 

 

 

y [ 80 (2/3)  + 85  + 89 ]  / [  y [( 2/3) + 1 + 1] ]       [cancel the y's ]

 

[ 80 (2/3) + 85 + 89]  [  8/3 ]

 

[ 160/3 + 85 + 89 ] /  [8/3] =

 

[ 160/3  + 255/3 + 267/3]  / [8/3]     multiply  top/bottom by 3

 

[160 + 255 + 267 ] / 8  = 

 

85.25  = average for all three classes

 

 

cool cool cool

14-oct-2018