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Intern  Joined: 28 Jun 2008
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At a certain university, the ratio of the number of teaching assistants to the number of students in any course must always be greater than 3:80. At this university , what is the maximum number of students possible in a course that has 5 teaching assistants?

A. 130
B. 131
C. 132
D. 133
E. 134

Originally posted by scorpio7 on 04 Jun 2009, 21:11.
Last edited by Bunuel on 13 Mar 2012, 00:35, edited 1 time in total.
Edited the question and added the OA
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spyguy wrote:
can someone explain in further detail the relationship between the teaching assistants to the number of students in any course must always be greater than 3:80 and how to reason through this portion? I understand how to solve for x. Once I was at this point I think was stumped on which number to select and inevitably chose to round up. My rational being .33 of a student is not possible therefore it must represent the position of an entire student. Thoughts? Help?

At a certain university, the ratio of the number of teaching assistants to the number of students in any course must always be greater than 3:80. At this university, what is the maximum number of students possible in a course that has 5 teaching assistants?
A. 130
B. 131
C. 132
D. 133
E. 134

Given: $$\frac{assistants}{students}>\frac{3}{80}$$ --> $$assistants=5$$, so $$\frac{5}{s}>\frac{3}{80}$$ --> $$s_{max}=?$$

$$\frac{5}{s}>\frac{3}{80}$$ --> $$s<\frac{5*80}{3}\approx{133.3}$$ --> so $$s_{max}=133$$.

$$\frac{assistants}{students}>\frac{3}{80}$$ relationship means that if for example # of assistants is 3 then in order $$\frac{assistants}{students}>\frac{3}{80}$$ to be true then # of students must be less than 80 (so there must be less than 80 students per 3 assistants) on the other hand if # of students is for example 80 then the # of assistants must be more than 3 (so there must be more than 3 assistants per 80 students).

Hope it's clear.
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At a certain university, the ratio of the number of teaching assistants to the number of students in any course must always be greater than 3:80. At this university, what is the maximum number of students possible in a course that has 5 teaching assistants?
A. 130
B. 131
C. 132
D. 133
E. 134

Originally posted by satishreddy on 24 Oct 2010, 17:57.
Last edited by Bunuel on 01 Feb 2012, 14:24, edited 1 time in total.
Edited the question
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Manager  Joined: 12 Apr 2006
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Location: India
Re: problem solving question on ratios  [#permalink]

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Not sure whether this is the best possible way but just the way how I solve it.

Teaching Assistants = TA
Students = S

Let assume the ratio of TA/S = $$3/80$$ (Just putting aside the requirement it must be greater)

Let say x be the maximum no of students possible with 5 teaching assistants = $$3/80 = 5/x$$

$$x = 400/3 = 133.33$$. Now for ratio to be greater than $$3/80$$ reduce the denominator. So just rounded it to lowest integer as number of student can't be in decimal. The new ratio is $$5/133$$, which is less than $$3/80$$ thus, 133 is the maximum number of students possible.
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At a certain university, the ratio of the number of teaching assistants to the number of students in any course must always be greater than 3:80. At this university, what is the maximum number of students possible in a course that has 5 teaching assistants?
A. 130
B. 131
C. 132
D. 133
E. 134

Given: $$\frac{assistants}{students}>\frac{3}{80}$$ --> $$assistants=5$$, so $$\frac{5}{s}>\frac{3}{80}$$ --> $$s_{max}=?$$

$$\frac{5}{s}>\frac{3}{80}$$ --> $$s<\frac{5*80}{3}\approx{133.3}$$ --> so $$s_{max}=133$$.

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Re: problem solving question on ratios  [#permalink]

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can someone explain in further detail the relationship between the teaching assistants to the number of students in any course must always be greater than 3:80 and how to reason through this portion? I understand how to solve for x. Once I was at this point I think was stumped on which number to select and inevitably chose to round up. My rational being .33 of a student is not possible therefore it must represent the position of an entire student. Thoughts? Help?
Intern  Joined: 05 Nov 2010
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Bunuel,

That is very clear. Thanks for breaking it down like that as it is more clear in order to solve future problems.
Retired Moderator Joined: 20 Dec 2010
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A ratio of Teacher/Student > 3/80

In words;
3 Teachers teach < 80 students
1 teacher teaches < 80/3 students
5 teachers teach < (80/3)*5 students
5 teachers teach < 400/3 students
5 teachers teach < 133.33 students

Students can only be integers
5 teachers teach < 134 students

or a maximum of 133 Students.

Ans: "D"
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hi,

I solved it this way, any suggestions always welcome

c/b > 3/80 ( from question)

5/b > 3 / 80
(80 x 5 / 3) > b

This reduces to

133.3333 > b

So the number of burgers have to be less than 133.33 & as u dont get 0.33 burger in Mc Donalds Max burgers is 133

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Originally posted by boomtangboy on 12 Mar 2012, 23:42.
Last edited by boomtangboy on 12 Mar 2012, 23:55, edited 1 time in total.
Manager  Joined: 09 Jul 2013
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The question states that the ratio must always be greater than 3:80, not the number of students (or burgers). So when you calculate the ratio $$\frac{5}{x}>\frac{3}{80}$$, increasing the value of $$x$$ will decrease the ratio $$\frac{5}{x}$$, and decreasing the value of $$x$$ will increase the ratio $$\frac{5}{x}$$.

If you calculate the number of burgers to be 133.3, then decide whether to round up or down, understand what will happen to the ratio of $$\frac{5}{x}$$.

If $$\frac{5}{133.33}=\frac{3}{80}$$, and that is the minimum (because $$\frac{5}{x}$$ must always be greater than $$\frac{3}{80}$$), what happens if you round $$x$$ up to 134? Is $$\frac{5}{134}$$ > or < $$\frac{3}{80}$$?

As explained above, if you increase $$x$$ to 134, then the ratio $$\frac{5}{x}$$ is decreased, and it will be less than the minimum of $$\frac{3}{80}$$. If you round $$x$$ down to 133, then the ratio $$\frac{5}{x}$$ will increase, and you will not violate the condition that it must always be greater than $$\frac{3}{80}$$.

Looking at it another way, if we know that the ratio of assistants to students must always be greater than 3:80, then we know that for any given number of assistants, there is a maximum number of students allowed. For every assistant, a maximum of 26.66 students are allowed (80/3). So if there is 1 assistant and 27 students, that is too many. 26 is the maximum number of students allowed if there is only 1 assistant in order to keep the ratio greater than 3:80. Using the same logic, if there are 5 assistants, then the maximum number of students allowed is 133.33. If there were 134 students that would be more than the maximum, therefore the maximum number of students allowed is 133.

Does that help?

Cheers
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Re: At a certain university, the ratio of the number of teaching  [#permalink]

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scorpio7 wrote:
At a certain university, the ratio of the number of teaching assistants to the number of students in any course must always be greater than 3:80. At this university , what is the maximum number of students possible in a course that has 5 teaching assistants?

A. 130
B. 131
C. 132
D. 133
E. 134

$$\frac{Assistant}{Student} > \frac{3}{80}$$

$$\frac{5}{Student} > \frac{3}{80}$$

$$Student < \frac{(5*80)}{3}$$

$$Student < \frac{(400)}{3}$$

i.e. $$Student < 133.33$$

i.e. Maximum value of No. of students = 133

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At a certain university, the ratio of the number of teaching  [#permalink]

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Brute Force Method:

$$\frac{3}{80}$$ As we are looking to a similar ratio for 5 assistants instead of 3, convert the both numerator (3) and denominator (80) to multiple of 5 by multiplying with 5

$$\frac{3*5}{80*5}$$equivalent to

$$\frac{15}{400}$$

Now as we need ratio for 05 assistants; again divide both numerator and denominator with 3. Pay attention to denominator which we need to answer:

$$\frac{5}{133.33}$$ (Post division of both numerator and denominator with 03)

DONE

Maximum students could be 133 because if students 134 ratio would be less.

Hope it helps!!!!
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Re: At a certain university, the ratio of the number of teaching  [#permalink]

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satishreddy wrote:
At a certain university, the ratio of the number of teaching assistants to the number of students in any course must always be greater than 3:80. At this university, what is the maximum number of students possible in a course that has 5 teaching assistants?
A. 130
B. 131
C. 132
D. 133
E. 134

We are given that the ratio of the number of teaching assistants to the number of students in any course must always be greater than 3:80, and we need to determine the maximum number of students possible in a course that has 5 teaching assistants. Let’s use the following formula, in which t = 5 = the number of teaching assistants and s = the number of students.

t/s> 3/80

5/s > 3/80

400 > 3s

400/3 > s

133.33 > s

Since s must be a whole number, the largest possible value of s is 133.

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Re: At a certain university, the ratio of the number of teaching  [#permalink]

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Does anybody know how I can solve the Problem with a ratio box?
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Hello Bunuel,

As per your explanation here, then max should be A: 130 to be greater than 3/80. Is my thinking correct.

Bunuel wrote:
spyguy wrote:
can someone explain in further detail the relationship between the teaching assistants to the number of students in any course must always be greater than 3:80 and how to reason through this portion? I understand how to solve for x. Once I was at this point I think was stumped on which number to select and inevitably chose to round up. My rational being .33 of a student is not possible therefore it must represent the position of an entire student. Thoughts? Help?

At a certain university, the ratio of the number of teaching assistants to the number of students in any course must always be greater than 3:80. At this university, what is the maximum number of students possible in a course that has 5 teaching assistants?
A. 130
B. 131
C. 132
D. 133
E. 134

Given: $$\frac{assistants}{students}>\frac{3}{80}$$ --> $$assistants=5$$, so $$\frac{5}{s}>\frac{3}{80}$$ --> $$s_{max}=?$$

$$\frac{5}{s}>\frac{3}{80}$$ --> $$s<\frac{5*80}{3}\approx{133.3}$$ --> so $$s_{max}=133$$.

$$\frac{assistants}{students}>\frac{3}{80}$$ relationship means that if for example # of assistants is 3 then in order $$\frac{assistants}{students}>\frac{3}{80}$$ to be true then # of students must be less than 80 (so there must be less than 80 students per 3 assistants) on the other hand if # of students is for example 80 then the # of assistants must be more than 3 (so there must be more than 3 assistants per 80 students).

Hope it's clear.
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Hi Aaron,

Bunuel's explanation is, of course, correct. The answer is D.

Think of it this way:
If you have a ratio $$\frac{a}{b}$$, and you want to increase the value of that ratio, there are two ways you can do that. You can either increase the numerator or decrease the denominator. In our case, we want to choose the denominator so that the resulting ratio is less than the given value $$\frac{3}{80}$$. To do that we will have to increase the denominator as much as we can so that the ratio $$\frac{assistants}{students}$$ is still greater than $$\frac{3}{80}$$. But if we increase the denominator too much the ratio of assistants to students will drop below our limit of $$\frac{3}{80}$$.

Imagine a simpler scenario where the target ratio wasn't $$\frac{3}{80}$$, but instead $$\frac{5}{80}$$. In that case the greatest number of students that would be allowed and still maintain the ratio of assistants to students greater than $$\frac{5}{80}$$ would be 79. One more student and the ratio would be equal to, not greater than $$\frac{5}{80}$$.

In our problem, we want the maximum number of students such that the ratio $$\frac{assistants}{students}>\frac{3}{80}$$. You can solve this algebraically, as has been done above to find that students must be less than 133.3. The largest integer less than 133.3 is 133. If we decrease the students down to 130, then the ratio will indeed be larger than the target of $$\frac{3}{80}$$, but we still have space to add more students and still maintain the ratio greater than $$\frac{3}{80}$$.

I hope it helps!

aaronTgmaT wrote:
Hello Bunuel,

As per your explanation here, then max should be A: 130 to be greater than 3/80. Is my thinking correct.

Bunuel wrote:
spyguy wrote:
can someone explain in further detail the relationship between the teaching assistants to the number of students in any course must always be greater than 3:80 and how to reason through this portion? I understand how to solve for x. Once I was at this point I think was stumped on which number to select and inevitably chose to round up. My rational being .33 of a student is not possible therefore it must represent the position of an entire student. Thoughts? Help?

At a certain university, the ratio of the number of teaching assistants to the number of students in any course must always be greater than 3:80. At this university, what is the maximum number of students possible in a course that has 5 teaching assistants?
A. 130
B. 131
C. 132
D. 133
E. 134

Given: $$\frac{assistants}{students}>\frac{3}{80}$$ --> $$assistants=5$$, so $$\frac{5}{s}>\frac{3}{80}$$ --> $$s_{max}=?$$

$$\frac{5}{s}>\frac{3}{80}$$ --> $$s<\frac{5*80}{3}\approx{133.3}$$ --> so $$s_{max}=133$$.

$$\frac{assistants}{students}>\frac{3}{80}$$ relationship means that if for example # of assistants is 3 then in order $$\frac{assistants}{students}>\frac{3}{80}$$ to be true then # of students must be less than 80 (so there must be less than 80 students per 3 assistants) on the other hand if # of students is for example 80 then the # of assistants must be more than 3 (so there must be more than 3 assistants per 80 students).

Hope it's clear.

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At a certain university, the ratio of the number of teaching  [#permalink]

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Another way to do it using remainders:

80 students /3 professors = 26 Remainder 2/3, so each professor teaches 26 and 2/3 students.
(26 Remainder 2/3) * 5 = 130 + 10/3 or 133 and 1/3
Because students have to be an integer, the max number that can be taught is D) 133 At a certain university, the ratio of the number of teaching   [#permalink] 10 Jun 2019, 09:14
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