Joe is given five tests that all have a maximum score of . The mean of his test scores is 90. If all his tests have a different grade and all the grades are whole numbers, what is the lowest grade he could have scored?

We know that he took 5 tests, mean is equal 90 so we know that following equations will be true if x will be sum of 5 scores:
\( \frac{x}{5}=90 \)
\( x=450 \)
Also we have to remember that max points that he could get is 100 so to find the lowest score he could get we have to do equation like this 
\( \frac{100+y}{2}=90 \)
\( 100+y=180 \)
\( y=80 \)

If the Mean = 90 for 5 test values
it means that there was a variation of values between maximum i-e 100 and minimum that may be any number not known.
The possible minimum value is when 4 tests result in 100 and the 5th is unknown
it means
Mean     = [(4X 100)+x] / 5
        90 = (400+x) / 5
400 + x  = 90 X 5
         x  = 450 -400 = 50

So the minimum possible test result is 50.