Test Average Word Problem

Date: 02/20/2004 at 10:22:51
From: Rich
Subject: Difficult Word Problem

In a certain class there are more than 20 and fewer than 40 students.
On a recent test the average passing mark was 75.  The average failing 
mark was 48 and the class average was 66.  The teacher then raised 
every grade 5 points.  As a result the average passing mark became 
77.5 and the average failing mark became 45.  If 65 is the established 
minimum for passing, how many students had their grades changed from 
failing to passing?

Date: 02/20/2004 at 11:22:59
From: Doctor Greenie
Subject: Re: Difficult Word Problem

Hi, Rich --

Thanks for sending the interesting problem.  It was unlike any I had 
seen before; and in solving it I got to use one of my favorite 
mathematical "tricks".

Here's what we have....

Before changing the grades:

  average passing grade:  75
  class average:          66
  average failing grade:  48

After changing the grades:

  average passing grade:  77.5
  class average:          71
  average failing grade:  45

All of these numbers were given specifically in the problem except 
for the class average after the grades were raised; because every 
grade was raised by 5 points, the class average was raised by 5 
points.

What we do with these numbers is determine the ratio of passing 
grades to failing grades before and after.  To do this, we use my pet 
technique for solving "mixture" problems.

For the case before the grades were raised, the average passing grade 
is 9 points above the class average, and the average failing grade is 
18 points below the class average.  So what we have is a "mixture" 
problem, where a certain number of students with average grades 18 
points below average and another number of students with average 
grades 9 points above average combine to produce that average.  

With the average for one group twice as far from the overall average
as the average for the other group, the ratio of students in the two 
groups must be 2:1.  And since the overall average is closer to the 
average of the students who passed, the larger group of students is 
the group who passed.  So now we know that we have "2x" students 
who passed and "x" students who failed.  This means the total number 
of students in the class is 3x, so the total number of students is 
divisible by 3.

For the case after the grades have been raised, we perform the same 
analysis.  The passing average and failing average are, respectively, 
6.5 points above and 26 points below the overall average.  These two 
numbers are in the ratio 1:4; this means that now the ratio of 
students who passed to students who failed is 4:1; and this means the 
number of students in the class is divisible by 5.

So now we know that the number of students in the class is divisible 
by both 3 and 5, and has to be between 20 and 40.  Can you determine
how many students there are from that information?

Once you know the total number of students, can you use the pass:fail
ratios to determine how many students passed and failed with each of
the two scorings?  That will let you answer the actual question posed
in the problem.

I hope this helps.  Please write back if you have any further 
questions about any of this.

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