Problem Solving

BTGmoderatorDC wrote: ↑

Tue Mar 31, 2020 6:50 pm

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

Source: Veritas Prep

Given that the student-professor ratio at a certain university was 25 : 4, let's assume that there were 25x students and 4x professors, where x is a positive integer.

Thus, currently, there are (25x – 80) students and (4x – 17) professors.

New student-professor ratio = (25x – 80)/(4x – 17)

Note that there can be multiple answers to this question. This is a 'Could be true' type of question.

Also, note that the student-professor ratio was 25 : 4 = 6.25 : 1. And the ratio of expelled students to quit professors = 80 : 17 = 4.7 : 1. Since 6.25 > 4.7, it means that relatively fewer students were expelled than the professors who quit. This means that the new student-professor ratio would be greater than 6.25 : 1. Thus, Option A (63 : 12 = 5.2..: 1) and Option B (95 : 17 = 5. 5...: 1) cannot be answers.

Also, note that the denominator of the revised ratio 25x – 80)/(4x – 17) is (4x – 17), which is always odd; thus, Option D is ruled out. Thus, we have two options to choose from: Option C and E.

• Option C: (25x – 80)/(4x – 17) = 33/25; upon solving, we get x = 23. Since x = 23, a positive integer, this is one of the correct answers.

Though we got the answer, let's check Option E, too.

• Option E: (25x – 80)/(4x – 17) = 55/3; upon solving, we get x = 23. Since x = 4.75.., not a positive integer, this cannot be a correct answer.

The correct answer: C

Hope this helps!

-Jay
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BTGmoderatorDC wrote: ↑

Tue Mar 31, 2020 6:50 pm

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

Source: Veritas Prep

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.

Let’s 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