Finding an Average without the Individual Numbers

Date: 09/12/2006 at 15:15:58
From: Amanda
Subject: Average score of all 50 students

A total of 50 juniors and seniors were given a mathematics test.  The 
35 juniors attained an average score of 80 while the 15 seniors 
attained an average of 70.  What was the average score of all 50 
students who took the test?

I don't know where to begin.  It's hard to find the average when they 
don't give a set of numbers.

Date: 09/12/2006 at 16:03:51
From: Doctor Douglas
Subject: Re: Average score of all 50 students

Hi Amanda.

It's true that they didn't give the entire slate of scores, but you
still have enough information to find the average.

The average score that you want is the total score of all the students
divided by the total number of students.  Clearly the total number of
students is 50.  Now, what about the total score?

Can you figure out a way to obtain the total score of just the 35
juniors?  It is possible to figure out this aggregate total without
knowing all of the individual scores.  If you are able to figure this
out, I'm sure that you'll have no trouble finishing this problem.

Write back if you get stuck.
 
- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 09/12/2006 at 16:39:16
From: Amanda
Subject: Average score of all 50 students

I took the total score of all the students, which was 80 + 70 = 150. 
I took this total of 150 and divided it by the total number of 
students which was 50 and I got 3.  The average score of all 50
students is not 3, so I guess I need more help.

Date: 09/12/2006 at 16:53:01
From: Doctor Douglas
Subject: Re: Average score of all 50 students

Hi again, Amanda.

Let's consider just the 35 juniors.  How could they obtain an
average score of 80?  

Well, one way is if they each scored exactly 80:  80 + 80 + ... + 80
= 35 x 80 = 2800.  Another way is if one of them scored 2800 (because
of a huge amount of extra credit) and everyone else scored zero--that 
would still lead to an average score of 80, because 2800/35 = 80.  
Either of these ways is certainly a possibility--all that matters is
that the total score of the 35 juniors is 2800, so that the average
score of the 35 juniors is 80.

You want to find the total score of the seniors in a similar way, and 
add that to the total score of the juniors to get the total score of 
all 50 students.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/