Averaging Kelly's Test Scores

Date: 01/07/2002 at 19:12:55
From: Brittney
Subject: Kelly's test scores

Kelly's average score on four Spanish tests is 85.5. The average of 
her three highest scores is 87, and her two lowest scores are the same 
as each other. What is the average of her two highest test scores?

I am totally stumped. I really need help!

Brittney

Date: 01/07/2002 at 23:42:26
From: Doctor Pete
Subject: Re: Kelly's test scores

Hi,

Here's the way I went about solving this question. If the average 
score of her four tests is 85.5, what is the total number of points 
she received over the four tests?  We know it must be 4 x 85.5 = 342 
points, since the definition of "average" is the sum of all the 
scores, divided by the number of tests.  In other words,

     Average = Total / Number,

or equivalently,

     Average x Number = Total.

Similarly, we may ask the question "what is the sum of her three 
highest scores?"  We know the average is 87, and there were three such 
exams, hence the sum is 3 x 87 = 261 points. Since this total does not 
count her lowest test score, and the previous total of 342 points does 
count this lowest score, we must conclude that the lowest score is

     342 - 261 = 81 points.

Since we are also told that her lowest two scores are the same, we see 
that the total of her highest two scores must be

     342 - (2 x 81) = 180 points,

and since this is the sum of her two highest scores, the average of 
these two scores is half, or 90 points.

To summarize, we can organize the solution into a table:

     [Number of Tests] x [Average] = [Sum of Scores]
             4         x   85.5    =      342
             3         x   87      =      261
     -----------------------------------------------
             1(lowest)             =  342-261 = 81
             2(lowest) x   81      =      162
     -----------------------------------------------
             2(highest)x [Average] =  342-162 = 180

As you can see, we don't need algebra to solve this problem - although 
we could have described the situation using variables for her test 
scores, it is perhaps simpler to use the fundamental definition of 
averages to arrive at the correct answer. Also, we should be aware 
that the problem does not contain enough information to find each 
score, only the lowest two scores and the average of the top two 
scores.

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