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

Question

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

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

chetan2u · Accepted Answer

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.

AnthonyRitz · Answer

We can solve it faster like this:

Kurtosis · Answer

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.

Answer: C

JeffTargetTestPrep · Answer

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.

Answer: C

stonecold · Answer

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 :roll:

.

Sajjad1093 · Answer

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?

neelavb · Answer

I assume my method is right . It will take you not more than a minute to solve this.

We take 25x-80/4x-17= a/b
 => 5x-16/4x-17= a/5b
So the consequent will be a multiple of 5. There is only option C matching the same.

Posted from my mobile device

neelavb · Answer

You can see my answer to solve it in a minute or less

Posted from my mobile device

veron90 · Answer

Hi Bunuel,

Wanted to ask if this is the correct method of solving it by  approximation  :

OG Ratio S:P = 25:4

Decreasing 8 students by a factor of 1/3rd = 8 (approx)
Reducing 17 professors which is approximately a  1.4  factor in relating to the number of students

Thus the closest new ratio is:

33:5

Abir77 · Answer

Bunuel  please help me out in this solution. Do you think i can apply this solution for the same problems like this one ???

chetan2u · Answer

You cannot apply the rule to even this question. 
You have to be careful as we are dealing with variables, the 5 can easily get cancelled out. 
For example
Now (25x-80)/(4x-17)....let x=8, then (25*8-80)/(4*8-17)=120/15=24/3

chetan2u · Answer

No, that will not be correct always..
please refer the above post of mine.

MathRevolution · Answer

As the ratio is 25:4, hence the total number of people will be 29 or multiple of 29.

Let us assume that the total number of people was 145: Professors: 20 and Students: 125.

University expelled eighty students for acts of civil disobedience, seventeen professors quit in protest. Therefore,

Students: 125 - 80 = 45 and Professors: 20 - 17 = 3

New ratio of student: the professor is 45 : 3 or 15:1

Given ratio is 25:4 which is 6.xx . Option A has value 5.xx , Option B has value 5.5x - ELIMINATED

We have professors as 20(Even) - 17(Odd) = 3(odd). Hence, D (55:6) - ELIMINATED

15:1 is higher than 25:4 and as we increase the numbers of professors and students, the ratio 15:1 will get lower but will still be higher than 25:4.

Option E is greater than 15.xx- ELIMINATED

Answer C

kashyapmurthy · Answer

25:4 comes to around 6.25 : 1 (bear with this notion)

Since 80 students and 17 professors leave let's see what is the ratio there, 80:17 ~ 4... : 1

So on remove 80 students and 17 professor we are removing more professors that necessary to maintain the 6.25:1 ratio.

So the resulting proportion should be greater than 6.25:1

A)63:12 = 5.x:1 
 B)95:17 = 5.x:1 
 C) 33:5 > 6.25:1
D) 55:9 > 6.25:1 
 E) 55:3 is too high compared to 55:9 so eliminate

We can now backtrack with options C and D

samarpan.g28 · Answer

Form 25x−80/4x−17. 4x−17 will be odd so let's cancel A and D since the denominator will be odd. Between A, C, and E, (C) is correct.

bumpbot · Answer

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