Expert: Socrates - 6/21/2007

Question
Hi and thanks for your help with this word problem.

Professor Davis counts his midterm as 2/3 of the grade, and his final as 1/3 of the grade.   Jason scored only a 64 on the midterm.  What range of scores on the final exam would put Jason's average between 70 and 79 inconclusive?

This is what I have for the equation:
70  <  2/3x + 64 < 79
  ---------------------------------
               1/3
I know to subtract to subtract 64 from each side.  My instructor tells me to multply that by 3/2.  I get a weird range.  Please help!  

Answer
Let f be the score on the final. Jason's average will be

(2/3)(64) + (1/3)f

We want this average to be between 70 and 79

70  <= (2/3)(64) + (1/3)f <= 79

multiply through by by 3 to clear the fractions

210 <= 128 + f <= 237

subtract 128 from everything

82 <= f <= 109

The range is 82 to 109.  Of course, if the highest possible score is 100 , the range is 82 to 100 and 79 is not possible for Jason's final average. In this case the final average will be at most 76. To see this, just put in 100 for f in (2/3)(64) + (1/3)f