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The present ratio of students to teachers at a certain

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The present ratio of students to teachers at a certain [#permalink] New post   Updated on: 07 Dec 2012, 06:59
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The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of teachers?

(A) 5
(B) 8
(C) 10
(D) 12
(E) 15

Show Spoiler
This might be really embarrasing, since I normally score Q47 or higher, but this question really made me stumble.

What is wrong with my approach (I always use this approach on ratio questions and it normally works well):

Plug in E:

25:1 is the new ratio -> 1 = 1*15 = 15 teachers and 25 = 25*15 = 375 students

15 - 5 = 10 teachers before the increase
375 - 50 = 325 students before the increse

so the old ratio is 325 : 10 which is unequal to 30:1

Where is my mistake?

PS. No need to tell me others ways how to solve it. I am familiar with the 'equations ways'.


Cheers a lot and I am happy ti give away Kudos ;)
Lars
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Official Answer
E

Originally posted by Impenetrable on 26 Mar 2012, 03:48.
Last edited by Bunuel on 07 Dec 2012, 06:59, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Slightly embarrasing [#permalink] New post   26 Mar 2012, 04:02
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Impenetrable wrote:
The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of teachers ?

5
8
10
12
15


This might be really embarrasing, since I normally score Q47 or higher, but this question really made me stumble.

What is wrong with my approach (I always use this approach on ratio questions and it normally works well):

Plug in E:

25:1 is the new ratio -> 1 = 1*15 = 15 teachers and 25 = 25*15 = 375 students

15 - 5 = 10 teachers before the increase
375 - 50 = 325 students before the increse

so the old ratio is 325 : 10 which is unequal to 30:1

Where is my mistake?

PS. No need to tell me others ways how to solve it. I am familiar with the 'equations ways'.


Cheers a lot and I am happy ti give away Kudos ;)
Lars


The very first step is not correct: the present ratio of students to teachers is 30 to 1, not 25 to 1. So if you want to plug t=15 you should use 30:1 ratio, not 25:1.

Complete solution:

Given: \(\frac{s}{t}=\frac{30x}{x}\) and \(\frac{30x+50}{x+5}=\frac{25}{1}\). Solve \(\frac{30x+50}{x+5}=\frac{25}{1}\) for \(x\) --> \(x=15\) --> \(t=x=15\).

Answer: E.

Hope it helps.
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Re: The present ratio of students to teachers at a certain [#permalink] New post   26 Mar 2012, 04:16
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+1 E

Here's what I did
s/t = 30/1 => s=30T

S+50/t+5 = 25/1

Substitute S=30T in the above equation

30t+50 = 25t+125

5t = 75
t=15

I hope this helps. By the way I was facing the same situation & it helps if you just take a break. I couldnt even recognise simple s v errors :-p
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Re: The present ratio of students to teachers at a certain [#permalink] New post   10 Jan 2021, 07:15
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Impenetrable wrote:
The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of teachers?

(A) 5
(B) 8
(C) 10
(D) 12
(E) 15


I like to use two variables for these "changing ratios" questions.

The PRESENT ratio of students to teachers at a certain school is 30 to 1
Let S = # of students PRESENTLY
Let T = # of teachers PRESENTLY
We can write: S/T = 30/1
Cross multiply to get: (1)(S) = (30)(T)
Simplify: S = 30T

If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1
In this scenario...
S + 50 = # of students
T + 5 = # of teachers
So, we get: (S + 50)/(T + 5) = 25/1
Cross multiply to get: (S + 50)(1) = (25)(T + 5)
Expand: S + 50 = 25T + 125
Simplify to get: S = 25T + 75

We now have 2 equations with 2 variables:
S = 30T
S = 25T + 75

To solve, replace S in the bottom equation with 30T
We get: 30T = 25T + 75
So: 5T = 75
Solve: T = 15
So, there are PRESENTLY 15 teachers

Answer: E

Cheers,
Brent
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Re: The present ratio of students to teachers at a certain [#permalink] New post   26 Mar 2012, 04:14
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How stupid of me. I should slap myself.


Thanks a lot! ;)
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The present ratio of students to teachers at a certain [#permalink] New post   02 Aug 2014, 04:23
Hello,

I am referring to Bunuel's solution. Can somebody explain why we are setting up the equation using a variable for the first ratio (30x/1x), but not for the second one (25x/1x). This is what I did wrong when I tried to solve the problem.

Thanks,

P.
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Re: The present ratio of students to teachers at a certain [#permalink] New post   02 Aug 2014, 10:05
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Pasqualo wrote:
Hello,

I am referring to Bunuel's solution. Can somebody explain why we are setting up the equation using a variable for the first ratio (30x/1x), but not for the second one (25x/1x). This is what I did wrong when I tried to solve the problem.

Thanks,

P.


If you write the first ratio as 30x/x, then you cannot write the second one as 25x/x, because x's there are not the same. You could write it as 25y/y though.

Does this make sense?
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Re: The present ratio of students to teachers at a certain [#permalink] New post   02 Aug 2014, 10:32
Thanks for the quick answer Bunuel. I understand that the two ratios are different and should be translated into two different variables (say x and y), but how do we solve for two variables if we do that?

e.g,: 30x + 50 = 25 y (x+5)

I hope I am making sense here. Basically, I don't understand why we use the first ratio with a variable and the second one as a normal numeric value. By the way, thanks for your amazing work on this forum.

Cheers
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The present ratio of students to teachers at a certain [#permalink] New post   02 Aug 2014, 10:44
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Pasqualo wrote:
Thanks for the quick answer Bunuel. I understand that the two ratios are different and should be translated into two different variables (say x and y), but how do we solve for two variables if we do that?

e.g,: 30x + 50 = 25 y (x+5)

I hope I am making sense here. Basically, I don't understand why we use the first ratio with a variable and the second one as a normal numeric value. By the way, thanks for your amazing work on this forum.

Cheers


If you use two variables, the second one will be simply reduced, that's why we don't need it.

\(\frac{30x+50}{x+5}=\frac{25y}{y}\) --> reduce by y: \(\frac{30x+50}{x+5}=\frac{25}{1}\) --> solve for x.

Hope it's clear.
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Re: The present ratio of students to teachers at a certain [#permalink] New post   02 Aug 2014, 10:56
Great, now it's crystal clear. Thanks again
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Re: The present ratio of students to teachers at a certain [#permalink] New post   06 Aug 2014, 22:19
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\(\frac{s}{t} = \frac{30}{1}\)

\(\frac{s+50}{t+5} = \frac{25}{1}\)

Placing s=30t in the above equation

30t+50 = 25t+125

5t = 75

t=15

Answer = E
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Re: The present ratio of students to teachers at a certain [#permalink] New post   10 Aug 2015, 13:01
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Hi, is there a quicker way to do this like using Alligation or something that would not involve setting up algebraic equations?
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Re: The present ratio of students to teachers at a certain [#permalink] New post   10 Aug 2015, 13:09
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ggurface wrote:
Hi, is there a quicker way to do this like using Alligation or something that would not involve setting up algebraic equations?


You can try plugging in the values from the options.

Lets say T = 10 ---> S = 300 ---> 350/15 = a bit less than 25 (A and B will only make this ratio lesser, eliminate A and B)

Go for T = 12 --> S = 30*12 = 360 ---> 360+50 / 12+5 = less than 25. Either you can now go further with T = 15 or mark E as the answer .

Check: T = 15 ---> S = 30*15=450 ---> 450+50/20 = 25.
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Re: The present ratio of students to teachers at a certain [#permalink] New post   09 Jun 2016, 14:10
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Impenetrable wrote:
The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of teachers?

(A) 5
(B) 8
(C) 10
(D) 12
(E) 15


We are given that the ratio of students to teacher is 30 to 1. We can rewrite this using variable multipliers.

students : teachers = 30x : x

We are next given that student enrollment increases by 50 and the number of teachers increases by 5. With this change the new ratio becomes 25 to 1. We can put all this into an equation:

Students/Teachers  25/1 = (30x + 50)/(x + 5)

If we cross multiply we have:

25(x + 5) = 30x + 50

25x + 125 = 30x + 50

75 = 5x

15 = x

Since x is the present number of teachers, currently there are 15 teachers.

Answer E.
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Re: The present ratio of students to teachers at a certain [#permalink] New post   02 Jan 2020, 21:45
Hi VeritasKarishma

Why can't we use an approach similar to like this used on another ratio related question ?

\(https://gmatclub.com/forum/the-number-of-stamps-that-kaye-and-alberto-had-were-in-the-98116.html#p1640044\)
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The present ratio of students to teachers at a certain [#permalink] New post   03 Jan 2020, 00:01
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Impenetrable wrote:
The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of teachers?

(A) 5
(B) 8
(C) 10
(D) 12
(E) 15

Show Spoiler
This might be really embarrasing, since I normally score Q47 or higher, but this question really made me stumble.

What is wrong with my approach (I always use this approach on ratio questions and it normally works well):

Plug in E:

25:1 is the new ratio -> 1 = 1*15 = 15 teachers and 25 = 25*15 = 375 students

15 - 5 = 10 teachers before the increase
375 - 50 = 325 students before the increse

so the old ratio is 325 : 10 which is unequal to 30:1

Where is my mistake?

PS. No need to tell me others ways how to solve it. I am familiar with the 'equations ways'.


Cheers a lot and I am happy ti give away Kudos ;)
Lars


Saumya2403:

You can use weighted average here but the figures are not conducive to that method.

A1 = 1/31 - Concentration of teachers in the group
A2 = 5/55 = 1/11 - Concentration of teachers in the added group
Aavg = 1/26 - Concentration of teachers in the combined group

w1/w2 = (1/11 - 1/26) / (1/26 - 1/31) = 15*31/5*11 = 93/11

Ratio of total group earlier : group added = 93 : 11 = x : 55
So there were 465 people earlier out of which 1/31 were teachers i.e. (1/31)*465 = 15 teachers

Answer (E)
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Re: The present ratio of students to teachers at a certain [#permalink] New post   03 Jan 2020, 00:06
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jabhatta@umail.iu.edu wrote:
Hi VeritasKarishma

Why can't we use an approach similar to like this used on another ratio related question ?

\(https://gmatclub.com/forum/the-number-of-stamps-that-kaye-and-alberto-had-were-in-the-98116.html#p1640044\)


We can't use the LCM method here because the number in the group is different in the two cases. Teachers and students were added. So it is like 2 mixtures were added to give an avg.
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Re: The present ratio of students to teachers at a certain [#permalink] New post   17 Feb 2020, 13:58
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Re: The present ratio of students to teachers at a certain [#permalink] New post   12 Dec 2020, 18:37
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Video solution from Quant Reasoning:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
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Re: The present ratio of students to teachers at a certain [#permalink] New post   28 Apr 2021, 04:06
The difficulty level of this question here on GMAT Club is low (15%). However, on the Official Guide (through the APP efficient learning) is categorized as a hard question.

I don't understand - how is such a huge discrepancy possible?

Thank you a lot in advance!
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Re: The present ratio of students to teachers at a certain [#permalink]
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