60% of all students in a class commute by public transportation. If 30

Lets say there are 10 students in the class. (A B C D E F G H I J)
60% of these students travel by Public Transportation. i.e. 6 students.
30% of students aspire to be scientists i.e. 3 students.

Extreme Case I (No overlap)
A B C D E F travel by Public Transportation. G H I aspire to be scientists. Remaining 1 out of 10 i.e. 10% .
The percentage of students who do not travel by Public Transportation and who do not aspire to be scientists will be not less than 10%.

Extreme Case II (Complete overlap)
The three students who aspire to be scientists fall in the list of students who travel by Public Transportation
i.e. lets say A B C D E F travel by Public Transportation and A B C aspire to be scientists. So remaining will be 4 out of 10. i.e. 40 %
The percentage of students who do not travel by Public Transportation and who do not aspire to be scientists will be not be greater than 40%.

So the % of students who do not travel by Public Transportation and who do not aspire to be scientists will be between 10% -40 %
And there is only one answer in the choices. 30%

Answer B
Ambarish

Let x represent students who do not commute by public transportation and do not aspire to be scientists.
And assume total no of students is 100.
40-x<30
This implies x>10 i.e., answer is more than 10%.
As per the table values x should be less than 40
Therefore only Statement 2 can give possible value.
Option B is correct.

The correct answer is Option B

The given information can be tabulated as below:



To answer this question, we need to know the percentage range of X.

The maximum value of X is 40% (because for X > 40%, z becomes negative, and a negative value is not allowed for z since z denotes the number of a category of students)

Let’s now find the minimum value of X.

We can write:
Y + X = 70%
So, X = 70% - Y
The maximum value of Y is 60%
This means, the minimum possible value of X is 10%

Thus, X can have values between 10% - 40% of all students, inclusive.
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60% of the students commute by public transportation, so 40% don't. So the students don't commute by public transportation can't be 60%. Eliminate the third option.
If all students who don't commute by public transportation want to be scientists, then 10% don't commute by public transportation or want to be scientists. So at least 10% qualify. Eliminate the first option.

60% of all students in a class commute by public transportation. If 30% of the students aspire to be scientists, which of the following can be the percentage of students who do not commute by public transportation and do not aspire to be scientists?
I. 5%
II. 30%
III. 60%

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II and III

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This question can easily be done with the help of a double matrix method, taking the total number of students as 60 (LCM of 30 and 60)

Try to substitute different possible values, 5% of 60, 30% of 60 and 60% of 60 and we can see that only II satisfies the double matrix condition.

Therefore, answer to this should be II only.

This question can be solved easily by using Set Grid
Scientist Non Scientist Total
Public 60
Non Public X 40
Total 30 70 100

Now, lets check all possible value of X. This cannot be greater than 40, so 60% is gone right away.
Lets check for x=5%. If X=5%, the Non Public scientist would be 35%, which violates the 30% total sum rule. So, this way we can rule out 5%.

Lets check for x=30%. This fits well in our grid and is the only option. Hence answer choice (B)

kql5112, askhere, anewbeginning, Mechmeera: You all answered this question right! Great job done with the analysis!

Mechmeera: Just a slight correction in the solution posted by you:

You wrote 40 - x < 30

It should actually be 40 - x =< (less than or equal to) 30%

Do you see why the 'equal to' case is also possible?

Similarly, the correct upper limit for x will be: x > = 40%

Clarity about this nuance would be crucial to getting the question right, if the values given in options I, II and III were, say:
I. 5%
II. 10%
III. 30%

Hope this was useful!

Best Regards

Japinder
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Thank You EgmatQuantExpert for the question(s)

Yes you are right.
Thank You EgmatQuantExpert for mentioning the point.

Very nice explanation

Thanks

A: students that take public transportation
B: students that want to be scientists

|A|=.60
|B|=.30

We want to find the possible values of |A-complement,B-complement|
By DeMorgan's law

(A-complement,B-complement)=(AUB)-complement

The max value of (AUB)-complement is when AUB is minimized
AUB is minimized when B is a subset of A.
AUB=A
Therefore (AUB)-complement=A-complement
|A-complement|=.4

So max value is 0.4


The min value of (AUB)-complement is when AUB is maximized
AUB is maximized when A and B are disjoint.
In such a case |AUB|=|A|+|B|=.9
Therefore |(AUB)-complement|=0.1

So the min value is 0.1

Only option II satisfies these two conditions.



Final Answer: B

60% of all students commute by public transportation (P), so 40% do not commute by public transportation (NP). 30% of all students aspire to be scientists (S), so 70% do not aspire to be scientists (NS).

The maximum possible percentage of students who NP&NS is:

max(NP&NS) = min(NP,NS) = min(40%,70%) = 40%

The minimum possible percentage of students who NP&NS is:

min(NP&NS) = NP – max(NP&S) = NP – min(NP,S) = 40% - min(40%,30%)

min(NP&NS) = 40% - 30% = 10%

We see that the percentage in question must be between 10% and 40%, inclusive.

Therefore, the only valid option is statement II.

Answer: B
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