Problem Solving

AAPL wrote:Veritas Prep

The student-professor ratio at a certain university was 25:4, but when the university expelled eighty students for acts of civil disobedience, seventeen professors quit in protest. Which of the following could be the new student-professor ratio at the university?

A. 63:12
B. 95:17
C. 33:5
D. 55:6
E. 55:3

OA C

Since the original student to professor ratio was 25 : 4, we can let 25x be the number of students and 4x be the number of professors for some positive integer x. We are given that 80 students were expelled and 17 professors quit, and we need to determine the possible ratio after the students and professors left. We note that the number of students remaining at school is now (25x - 80), and the number of professors remaining is (4x - 17). Thus, we need to determine which one of the given choices can be the value of (25x - 80)/(4x - 17) for some positive integer x.

There is no shortcut to solve this problem; we must consider each answer choice.

A. Is there a positive integer x such that (25x - 80)/(4x - 17) = 63/12 ?

12(25x) - 12(80) = 63(4x) - 63(17)

300x - 960 = 252x - 1071

48x = -111

x = -111/48

Since x can't be negative or a fraction, choice A is not correct.

B. Is there a positive integer x such that (25x - 80)/(4x - 17) = 95/17 ?

17(25x) - 17(80) = 95(4x) - 95(17)

425x - 1360 = 380x - 1615

45x = -255

x = -255/45 = -51/9

Again, since x can't be negative or a fraction, choice B is not correct.

C. Is there a positive integer x such that (25x - 80)/(4x -17) = 33/5 ?

5(25x) - 5(80) = 33(4x) - 33(17)

125x - 400 = 132x - 561

161 = 7x

x = 23

We see that it is possible to have a ratio of 33/5 after 80 students and 17 professors leave the university.

Answer: C