Inside your loop you're reading into $name instead of $studentname. I'm not sure how your sample output still manages to print the correct names, since your printf uses $studentname as well.

You're also forgetting about the line terminators. The first time you'll correctly read the name and scores. The second time your name will actually contain the line feed from the first line followed by the first 19 characters of the second name (or 18 if it's CR/LF instead of just CR)

For each subsequent line things will keep shifting. That's probably why you never find the "END".

Argh...I put in the wrong code up here, then. I've been experimenting so much, and nothing works.
Even with that fixed, it still doesn't work. I'll change the code. The varnames, anyway...Line terminators? -_-

Try adding <INPUT>; after both of these lines

read (INPUT, $test4, 3);
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That will read/skip the linefeed which would otherwise remain on each line.

(And also fix $name --> $studentname, of course.)

There are many things "wrong", or at least sub-optimum. First, always use strictures (use strict; use warnings;). Those will pick up a whole slew of errors related to typos and brain farts.

Use the three parameter version of open and lexical variables (open my $inFile, ...) for file handles.

Use indentation to show the block structure of your code:

...

while (...) {
    ...
}

...
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You might like to point your teacher to PerlMonks. If your code is a faithful rendition of your teacher's style then the advice above applies to your teacher too!

# !perl/bin/perl -w
#Lab 7a
#XXXX XXXX  #My full name...Yeah.
#Calculates grade averages

open (INPUT, "Tscore.dat") || die "Cannot open input file.\n";
open (OUTPUT, ">L7AOUT.txt") || die "Cannot open file.\n";

print OUTPUT "      CLASS TEST AVERAGES\n\n";
print OUTPUT "                          TEST\n";
print OUTPUT "NAME OF STUDENT         AVERAGE\n\n";

read (INPUT, $studentname, 20);
read (INPUT, $test1, 3);
read (INPUT, $test2, 3);
read (INPUT, $test3, 3);
read (INPUT, $test4, 3);
read (INPUT, $eor, 1);
<INPUT>;

$studentname3 = uc(substr ($studentname, 0, 3));

while ($studentname3 ne "END")
{
$avg = ($test1 + $test2 + $test3 + $test4) / 4;
$studnum = $studnum + 1;
$totavg = $totavg + $avg;
printf OUTPUT ("%-24.20s %6.2f", $studentname, $avg);
print OUTPUT "\n\n";
read (INPUT, $studentname, 20);
read (INPUT, $test1, 3);
read (INPUT, $test2, 3);
read (INPUT, $test3, 3);
read (INPUT, $test4, 3);
read (INPUT, $eor, 1);
<INPUT>;

$studentname3 = uc(substr ($studentname, 0, 3));
}
$totavg = $totavg / $studnum;
print "\n\n";
printf OUTPUT ("%-24.22s %6.2f", $studnum, $totavg);
print "Program completed Successfully.";
close INPUT;
close OUTPUT;
exit 0;
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