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# Of all the students in a certain dormitory, 1/2 are first-year student

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Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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27 Dec 2009, 10:47
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Of all the students in a certain dormitory, 1/2 are first-year students and the rest are second-year students. If 4/5 of the first-year students have not declared a major and if the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major, what fraction of all the students in the dormitory are second-year students who have not declared a major?

A. 1/15
B. 1/5
C. 4/15
D. 1/3
E. 2/5
[Reveal] Spoiler: OA

Last edited by Bunuel on 22 Aug 2017, 11:43, edited 3 times in total.
Renamed the topic and edited the question.

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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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27 Dec 2009, 14:31
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I think you dropped a fraction when you typed/copied this in. Working backward from the spoiler answer, I'm thinking the second sentence should read as follows? Please confirm.

"If 4/5 of the first-year students have not declared a major ..."
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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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27 Dec 2009, 18:09
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msand wrote:
Of all the students in a certain dormitory, 1/2 are first-year students and the rest are second-year students. If 4/5 of the first-year students have not declared a major and if the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major, what fraction of all the students in the dormitory are second-year students who have not declared a major?

A. 1/15
B. 1/5
C. 4/15
D. 1/3
E. 2/5

OA :
[Reveal] Spoiler:
B

First-year=Second-year=x --> all students $$2x$$;

$$\frac{4}{5}x$$ of the first-year students have not declared a major --> $$\frac{1}{5}x$$ declared;

Second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major =$$\frac{3}{5}x$$ --> who have not $$\frac{2}{5}x$$;

Hence: $$\frac{\frac{2}{5}x}{2x}=\frac{1}{5}$$

Answer: B.
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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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27 Dec 2009, 22:10
tot students=x

1st year student=x/2----> non majaor=4/5(x/2)-----> maj=1/5(x/2)

2nd year student=x/2---->maj=3(1/5(x/2))=3/10(x)--->non major=x/2-3/10(x)=1/5(x)

hence 1/5
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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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12 Aug 2011, 15:23
I think the answer choices you have provided because the answer cannot be greater than one.

Here is an easy way to think about this is by picking "smart" numbers.
Let's pick 50 for the first half of students who are 1st years and 50 for the other half who are 2nd year.
We know that 4/5 of the 1st years have not declared a major so that gives us 50*4/5=40, so 10 have declared a major.
We also know that the number of 2nd year that have not declared a major = 3 times the # of 1st year who have declared a major. This gives us 3*10=30

The question asks for fraction of 2nd years who have not declared a major over the total students.
30 have declared a major out of the 2nd year so 20 have not. Since we have 100 students (50+50), the answer is 20/100 or 1/5.

You can run a search on the web and you will find this question without the correct answer choices.

Thanks

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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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30 Jan 2013, 07:25
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You can also do this problem sans algebra (in your head):

100 Students Total:

1st yr = 50 students
4/5x50 = 40 = No Major
Therefore: 10 = Major

2nd yr = 50 students
10x3 = 30 = Major
Therefore: 20 = No Major
20/100 = 1/5 = B
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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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08 Jun 2013, 07:34
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In this kind of problem table structure is the weapon for me...

Let 2X would be total student which revealed that 1 yrs student is X and 2nd Yr is X as well.

No form the table

MAJOR NO MAJOR Total Student
1st yr Student x/5 4x/5 x
2nd yr Student 3x/5 2x/5 x
Total Student 4x/5 6x/5 2x

From the table 2nd yr student with major is 2x/5 so the answer would be =2x/5/2x =>1/5 Ans

Rgds
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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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01 Sep 2014, 19:09
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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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03 Sep 2014, 02:26
First ............................................. Second ........................................ Total

x ........................................................ x ................................................ 2x

Major........... NotMajor ................... Major ............ NotMajor ..........

$$\frac{4x}{5}$$ ................ $$\frac{x}{5}$$ ........................ $$\frac{3x}{5}$$ ................... $$\frac{2x}{5}$$ ...................... 2x

Fraction $$= \frac{\frac{2x}{5}}{2x} = \frac{1}{5}$$

Answer = B
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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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28 Apr 2016, 23:06
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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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Attached is a visual that should help.

FY = First Year
SY = Second Year
NDM = Not Decided on Major
YDM = Yes Decided on Major
SYY = Senior Year Yes
SYN = Senior Year No

Attachments

Screen Shot 2016-05-05 at 6.59.42 PM.png [ 68.47 KiB | Viewed 2378 times ]

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Last edited by mcelroytutoring on 07 May 2016, 21:08, edited 6 times in total.

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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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05 May 2016, 21:38
mojorising800 wrote:
Of all the students in a certain dormitory, 1/2 are first-year students and the rest are second-year students. If 4/5 of the first-year students have not declared a major and if the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major, what fraction of all the students in the dormitory are second-year students who have not declared a major?

A. 151
B. 51
C. 154
D. 31
E. 52

Say there are in all 100 students. Use a matrix:
50 are first year and 50 are second year students.

........................Major Dec................Major not dec..............Total
First Year............................................................................50
Second Yr...........................................................................50
Total

4/5 * 50 = 40 not declared major first year students.

........................Major Dec................Major not dec..............Total
First Year................10..................................40....................50
Second Yr...........................................................................50
Total

second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major

........................Major Dec................Major not dec..............Total
First Year................10..................................40.....................50
Second Yr...............30..................................20......................50
Total

So second year students who've not declared a major is 20/100 = 1/5

Answer (B)
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Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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22 Aug 2017, 11:32
This problem is 99 in og 2017.

T = total students in dorm
4/5 first year no declared major = 4/5(.5T) = .4T = # first year no major
.5T - .4T = .1T = 1st year w/ major
.1T x 3 = .3T
2nd year no major = .5T - .3T = .2T = 2nd no declared
.2T does not equal 2/5!

.2T / T = .2 = 1/5

Last edited by CyberStein on 07 Oct 2017, 07:33, edited 1 time in total.

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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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22 Aug 2017, 11:45
CyberStein wrote:
msand wrote:
Of all the students in a certain dormitory, 1/2 are first-year students and the rest are second-year students. If 4/5 of the first-year students have not declared a major and if the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major, what fraction of all the students in the dormitory are second-year students who have not declared a major?

A. 1/15
B. 1/5
C. 4/15
D. 1/3
E. 2/5

As other people have mentioned, there are a few critical typos.

This problem is Q 99 from OG 2017.

The real answer is E, not B.

Here is my solution:
1/2 = first year
4/5 = first year no major; therefore, 5/5 (total first year) - 4/5 (first year no major) = 1/5 First year with major
1/5 = first year with major

"second year students who have declared a major is 3 times the fraction of first year who have declared a major"

Given this: 3/1 x 1/5 = 3/5 Second Year who have declared major.

But the question asks for the fraction of students who have not declared a major. Since we now know how many second year majors, we can answer the question by finding the difference between total second year and second year with major.

5/5 (total second year) - 3/5 (Second Year with major) = 2/5 (Second year students without a major)

(E) 2/5 is the answer

Yes, this is an Official Guide question and the Official Answer is B, 1/5, NOT E. There are several solutions from experts above, so if interested you can check them.
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Re: Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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22 Aug 2017, 16:05
msand wrote:
Of all the students in a certain dormitory, 1/2 are first-year students and the rest are second-year students. If 4/5 of the first-year students have not declared a major and if the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major, what fraction of all the students in the dormitory are second-year students who have not declared a major?

A. 1/15
B. 1/5
C. 4/15
D. 1/3
E. 2/5

Attachment:

matrix111.png [ 17.63 KiB | Viewed 580 times ]

Double matrix works well here, at least for me. I assumed 60 total students.

1. Number of dormitory students = 60. Half are first-year students = 30. So the other half, second-year students = 30

2. $$\frac{4}{5}$$ of first-year students have NOT declared a major: $$\frac{4}{5}$$ * 30 = 24

3. First-year students who HAVE declared a major = 30 total - 24 not = 6 who have

4. The fraction of second-year students who have declared a major is three times the fraction of first-year students who have declared a major. First-year declared = 6. Second-year declared = 3 * 6 = 18

5. Second-year students who have NOT declared a major: 30 total second-year students - 18 who have declared = 12 second-year students who have not

6. What fraction of ALL students are second-year students who have not declared a major? $$\frac{12}{60} = \frac{1}{5}$$

Answer B

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Of all the students in a certain dormitory, 1/2 are first-year student [#permalink]

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23 Aug 2017, 00:45
msand wrote:
Of all the students in a certain dormitory, 1/2 are first-year students and the rest are second-year students. If 4/5 of the first-year students have not declared a major and if the fraction of second-year students who have declared a major is 3 times the fraction of first-year students who have declared a major, what fraction of all the students in the dormitory are second-year students who have not declared a major?

A. 1/15
B. 1/5
C. 4/15
D. 1/3
E. 2/5

This question is pretty easy if we assume numbers. Used numbers because the question does not have fixed values and asks for percentages.

Assuming the total number of students in the dormitory to be 100

Given 1/2 are 1st year and other 1/2 are 2nd year. Hence 50 students each are in either year.

Given 1st year (Not decided Major)= 1stNDM=4/5th of 1st year=40

Hence 1st year (Decided Major)= 1stDM=10

Given 2nd year (Decided Major)= 2ndDM=3x1stDM=30

Hence 2nd year (Not decided Major)= 2ndNDM=100-(40+10+30)=20

Fraction of 2nd year students who havent declared a major = 20/100=1/5

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