GATE CSE 2024 | Set 2 | GA Question: 3

Question

​​​​In an engineering college of $10,000$ students, $1,500$ like neither their core branches nor other branches. The number of students who like their core branches is $1 / 4^{\text {th }}$ of the number of students who like other branches. The number of students who like both their core and other branches is $500$.

The number of students who like their core branches is

• A. $1,800$
• B. $3,500$
• C. $1,600$
• D. $1,500$

Comments

I think the answer should be 1600

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Same mistake I did, they are asking for who liked core branches, not who liked "ONLY" core branches

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Yes correct . they didn't asked only branch.

I also did this wrong.BTW should I take down my above comment ?.

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keep it so who visit later can get if they also do silly mistake like it.

Answer 1

Let $\Omega, C, O$ be the set of total students(universal set), students who like their core branch and students who like other branches respectively.

$\underline{\text{Given:}}\ \ \ |\Omega| = 10000,\ \ | C^\complement \cap O^\complement | = 1500,\ \ |O| = 4|C|,\ \ |C \cap O| = 500$

$\underline{\text{To find:}}\ \ |C| = \ ?$

We know that,

$\begin{align*}
|C \cup O| &= |\Omega| - |(C \cup O)^\complement| \\
|C \cup O| &= |\Omega| - | C^\complement \cap O^\complement | && \text{(De Morgan's law)} \\
|C \cup O| &= 10000 - 1500 = 8500 \\
\end{align*}
$

Also,

$\begin{align*}
|C \cup O| &= |C| + |O| - |C \cap O| \\
|C| + |O| &= |C \cup O|  + |C \cap O| \\
5|C| &= 8500 + 500  = 9000 \\
|C| &= 9000/5 = 1800
\end{align*}
$

So, the number of students who like their core branches is $1800$.

$\bf \therefore Ans = A.$

Answer 2

Answer is 1800

Comments

Understood..

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Just tell me one thing if ans is 1800

students like core branch = 1800

Students like other branch = 1800 X 4 = 7200

Students who neither like core branch nor other = 1500

Now carefully observe total

1800+7200+1500 = 10500

How it can get possible if students are 10000

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(because total no of student who like either their or other's or both's core branches is 10,000-1500=8500) 

8500=1800 is who like their core branches + 7200 who like other's core - who like their core and other's core 

so 500 people like the both branches, U have concidered twice in u calcuation ..
did U understood?

Answer 3

1800

Comments

please check your ans may be wrong it will be 1600

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How come?

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Yes you are right

Total students= 10000

Students who don't like their core or other branches = 1500

No of students who like both branches= 500

Assume number of students who like other branches = X

there fore given that no of students who like their core branches are X/4

therefore X+X/4 = 8500

So X = 6400

so students who like their core branches are: X/4= 1600

So the answer is 1600

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You are overcounting at "x+x/4=8500".You need to remove the intersection (500) once since you have counted it twice (once in x and and once in x/4. This is inclusion-exclusion principle)