Applications tenure

Question

Professor Walt is up for tenure, and wishes to submit a portfolio of written student evaluations as evidence of his good teaching. He begins by grouping all the evaluations into four categories: good reviews, bad reviews (a typical one being "GET RID OF WALT! THE MAN CAN'T TEACH!"), mediocre reviews (such as "I suppose he's OK, given the general quality of teaching at this college"), and reviews left blank. When he tallies up the piles, Walt gets a little worried: There are 280 more bad reviews than good ones and only half as many blank reviews as bad ones. The good reviews and blank reviews together total 170. On an impulse, he decides to even up the piles a little by removing 280 of the bad reviews, and this leaves him with a total of 400 reviews of all types. How many of each category of reviews were there originally?

Answer 1

Let a = # of good reviews, b = # of bad, c = # of mediocre, d = # of blank.

 

Given:

 

b = a + 280

b = 2d

a + d = 170

a + b + c + d - 280 = 400  => a + b + c + d = 680

 

Let's solve for d in terms of a.

 

a + d = 170

=> d = 170 - a

 

Since b = a + 280 and b = 2d, we can set them equal to each other.

 

a + 280 = 2d

 

Substitute for d and solve for a.

 

a + 280 = 2(170 - a)

a + 280 = 340 - 2a

=> 3a = 60

=> a = 20

 

Now we can go back to our equation for d in terms of a and solve.

 

d = 170 - a

   = 170 - 20 = 150

 

Now we can use a given equation for b to solve. Either would do.

 

b = 2d

   = 2(150) = 300

 

 

Finally, use the final given equation, plug in all known variables, and solve for c.

 

a + b + c + d = 680

20 + 300 + c + 150 = 680

470 + c = 680

=> c = 210

 

So we have 20 good reviews, 300 bad reviews, 210 mediocre reviews, and 150 blank reviews.