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### Averaging Kelly's Test Scores

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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/
```
Associated Topics:
High School Statistics
Middle School Statistics

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