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Math Expert V
Joined: 02 Sep 2009
Posts: 55668
The student-professor ratio at a certain university was 25:4, but when  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 35% (02:45) correct 65% (02:54) wrong based on 249 sessions

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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

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Math Expert V
Joined: 02 Aug 2009
Posts: 7757
Re: The student-professor ratio at a certain university was 25:4, but when  [#permalink]

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Bunuel wrote:
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

Hi...
Ratio is $$\frac{25}{4}$$...
If common factor is x, then the numbers become 25x and 4x...
If you take out 80 students and 17 instr, numbers become 25x-80 and 4x-17...

We can eliminate few choices straight way..
4x-17 will be always ODD, so choices with even ratio in denominator can be left out..
So A and D are out..

Let's check the remaining..
Start from centre as you would know if you have to move to a higher or lower ratio after checking the middle value.
So $$\frac{25x-80}{4x-17}=33/5..........(25x-80)*5=(4x-17)*33....125x-400=132x-561......7x=161....x=23$$
Since x is an integer value, this is our answer.
C

We can check for other values but x should not be integer value.
Let's check ONE
So $$\frac{25x-80}{4x-17}=\frac{55}{3}..........(25x-80)*3=(4x-17)*55....75x-240=220x-935......145x=695....$$
Since x is not an integer value, this is not our answer.
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Re: The student-professor ratio at a certain university was 25:4, but when  [#permalink]

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Backsolve:

A) (63x + 80)/(12x + 17) = 25/4
252x + 320 = 300x + 425 --> x is negative. Eliminate A.

B) (95x + 80)/(17x + 17) = 25/4
380x + 320 = 425x + 425 --> x is negative. Eliminate B.

C) (33x + 80)/(5x + 17) = 25/4
132x + 320 = 125x + 425
7x = 105 --> x = 15. Keep.

D) (55x + 80)/(6x + 17) = 25/4
220x + 320 = 150x + 425
70x = 105 --> x = 1.5. But x can't be a fraction as the number of students is an integer. Eliminate D.

E) (55x + 80)/(3x + 17) = 25/4
220x +320 = 75x + 425
145x = 105 --> x = 21/29 and the number of students will be a fraction. Eliminate E.

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Re: The student-professor ratio at a certain university was 25:4, but when  [#permalink]

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Bunuel wrote:
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

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. 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 go over 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.

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GRE 1: Q169 V154 Re: The student-professor ratio at a certain university was 25:4, but when  [#permalink]

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1
Extremely Lengthy Question.
No tricks in place.
My time => 3 minutes and 24 seconds.
Method used => Back-solving.

How can GMAT expect someone solve solve it under 2-2.5 minutes     .

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The student-professor ratio at a certain university was 25:4, but when  [#permalink]

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stonecold wrote:
Extremely Lengthy Question.
No tricks in place.
My time => 3 minutes and 24 seconds.
Method used => Back-solving.

How can GMAT expect someone solve solve it under 2-2.5 minutes     .

We can solve it faster like this:

Veritas Prep Official Solution wrote:
Trying to solve this problem algebraically would be a terrible slog. However, a bit of cleverness can avoid all that.

Begin by noting that the 80:17 ratio in which students and professors were subtracted is less than the initial 25:4 ratio. As a consequence, the final ratio should be greater than the initial ratio.

Since this ratio principle is lesser-known (but still testable), let's demonstrate with an example: Consider beginning with a 5:2 ratio and subtracting a smaller 2:1 ratio. If the numbers are literally 5 and 2 and we subtract literally 2 and 1, the resulting ratio will be 3:1 -- greater than the initial 5:2 = 2.5:1 ratio. Scaling up to 10:4 and then subtracting 2 and 1 would result in a ratio of 8:3≈2.67:1. Scaling up again to 15:6 would result in a final ratio of 13:5=2.6:1. And so forth.

This principle is enough to eliminate A and B, since the final ratio should be greater than the initial ratio in this problem and since 25:4 = 6.25 but 63:12 ≈ 5.1 and 95:17 ≈ 5.5.

Next, note that we are subtracting an odd number of professors (17) from an even number of professors (4x). Thus the final number of professors must be odd. This eliminates D (and eliminates A for a second time).

We're left with C and E. To finish things off, let's start to scale up from literally 25 students and 4 professors and see what happens. Until the number of students exceeds 80 and the number of professors exceeds 17 the numbers will make no sense -- the numbers of students and professors have to remain positive after the change takes place. So the smallest possible initial numbers are 25∗5 = 125 students with 4∗5 = 20 professors (the professors were the limiting factor here). At these values, subtracting 80 students and 17 professors would give final values of 45 students and 3 professors, resulting in a student-professor ratio of 45:3, which is 15:1.

Finally, observe that as the initial numbers of students and professors continue to increase, the final ratio will fall and move closer and closer to the initial 25:4. After all, if the numbers of students and professors become very large, 80 students and 17 professors will become a "drop in the bucket" that makes barely any difference to the ratio.

We can see this numerically by looking at the next larger numbers of students and professors: 150 and 24, respectively. Subtracting 80 and 17 would give a new ratio of 70:7 = 10:1. Sure enough, this ratio would be less than the 15:1 ratio that we now recognize as the maximum but still greater than the 25:4 ratio that we now recognize as the minimum.

Since answer E features a 55:3 ratio that is greater than 15:1, we can eliminate E and conclude that the correct answer must be C.
Intern  B
Joined: 11 Mar 2017
Posts: 12
Re: The student-professor ratio at a certain university was 25:4, but when  [#permalink]

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Hi..

Thanks for the solution. just confused where you say 4x-17 is always ODD so we can cancel options with even denominator, but we are talking about the ratio here rather than actual values. for instance if 25x-80 is also odd then an ODD factor cancelling both the terms out could also have resulted in an even denominator as well?

chetan2u wrote:
Bunuel wrote:
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

Hi...
Ratio is $$\frac{25}{4}$$...
If common factor is x, then the numbers become 25x and 4x...
If you take out 80 students and 17 instr, numbers become 25x-80 and 4x-17...

We can eliminate few choices straight way..
4x-17 will be always ODD, so choices with even ratio in denominator can be left out..
So A and D are out..

Let's check the remaining..
Start from centre as you would know if you have to move to a higher or lower ratio after checking the middle value.
So $$\frac{25x-80}{4x-17}=33/5..........(25x-80)*5=(4x-17)*33....125x-400=132x-561......7x=161....x=23$$
Since x is an integer value, this is our answer.
C

We can check for other values but x should not be integer value.
Let's check ONE
So $$\frac{25x-80}{4x-17}=\frac{55}{3}..........(25x-80)*3=(4x-17)*55....75x-240=220x-935......145x=695....$$
Since x is not an integer value, this is not our answer. Re: The student-professor ratio at a certain university was 25:4, but when   [#permalink] 26 Oct 2018, 23:05
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