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A survey was conducted to find out how many people in a hous [#permalink]
19 Sep 2011, 09:00

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Difficulty:

65% (medium)

Question Stats:

48% (03:43) correct
52% (02:59) wrong based on 104 sessions

A survey was conducted to find out how many people in a housing colony of 144 residents could swim, dance and drive a car. It was found that the number of people who could not swim was 89, the number of people who could not dance was 100 and that the number of people who could not drive a car was 91. If the number of people who could do at least two of these things, was found to be 37 and the number of people who could do all these things was found to be 6, how many people could not do any of these things?

T=n(A)+n(B)+n(C)-n(Exactly two of the events)-2*n(All 3 Events)+n(None of the events)

T=144 n(A)=T-n(A')=144-89=55 n(B)=T-n(B')=144-100=44 n(C)=T-n(C')=144-91=53 n(Exactly two of the events)=n(At least 2 Events)-n(All 3 Events)=37-6=31 n(All 3 Events)=6

144=55+44+53-31-2*6+n(None of the events) n(None of the events)=144-55-44-53+31+12=35

I used to be confused with such kind of problems and what helped me is drawing Venn. If you follow this there is no need to remember any formula. One just need ot remember simple +/- and also take less than 2 min.

A survey was conducted to find out how many people in a housing colony of 144 residents could swim, dance and drive a car. It was found that the number of people who could not swim was 89, the number of people who could not dance was 100 and that the number of people who could not drive a car was 91. If the number of people who could do at least two of these things, was found to be 37 and the number of people who could do all these things was found to be 6, how many people could not do any of these things?

A) 17

B) 23

C) 29

D) 35

E) 50

I am not sure that stated official answer is correct one. or I am tricked by the language (which generally is the case in these kind of question). IMHO, the number of people who could do at least two of the things also includes 18 ( 3 *6 where 6 are the number of people who could all these things). My answer is coming as option B - 23. Can anybody help.

A survey was conducted to find out how many people in a housing colony of 144 residents could swim, dance and drive a car. It was found that the number of people who could not swim was 89, the number of people who could not dance was 100 and that the number of people who could not drive a car was 91. If the number of people who could do at least two of these things, was found to be 37 and the number of people who could do all these things was found to be 6, how many people could not do any of these things?

A) 17

B) 23

C) 29

D) 35

E) 50

I am not sure that stated official answer is correct one. or I am tricked by the language (which generally is the case in these kind of question). IMHO, the number of people who could do at least two of the things also includes 18 ( 3 *6 where 6 are the number of people who could all these things). My answer is coming as option B - 23. Can anybody help.

The number of people who can do at least two things includes 6 (number of people who can do all three), not 6*3.

Understand here that 37 is the number of people, not the number of instances. Hence 6 is not counted 3 times in 37. Out of 37 people, 18 people cannot do all three. Only 6 can do all three. So 31 can do exactly 2 things.

T=n(A)+n(B)+n(C)-n(Exactly two of the events)-2*n(All 3 Events)+n(None of the events)

T=144 n(A)=T-n(A')=144-89=55 n(B)=T-n(B')=144-100=44 n(C)=T-n(C')=144-91=53 n(Exactly two of the events)=n(At least 2 Events)-n(All 3 Events)=37-6=31 n(All 3 Events)=6

144=55+44+53-31-2*6+n(None of the events) n(None of the events)=144-55-44-53+31+12=35

Ans: "D"

But the stem says "at least 2", not exactly. so it should be the formula as follows:

T=n(A)+n(B)+n(C)-n(at least two of the events)+n(All 3 Events)+n(None of the events)

A survey was conducted to find out how many people in a housing colony of 144 residents could swim, dance and drive a car. It was found that the number of people who could not swim was 89, the number of people who could not dance was 100 and that the number of people who could not drive a car was 91. If the number of people who could do at least two of these things, was found to be 37 and the number of people who could do all these things was found to be 6, how many people could not do any of these things?

A) 17

B) 23

C) 29

D) 35

E) 50

I am not sure that stated official answer is correct one. or I am tricked by the language (which generally is the case in these kind of question). IMHO, the number of people who could do at least two of the things also includes 18 ( 3 *6 where 6 are the number of people who could all these things). My answer is coming as option B - 23. Can anybody help.

The number of people who can do at least two things includes 6 (number of people who can do all three), not 6*3.

Understand here that 37 is the number of people, not the number of instances. Hence 6 is not counted 3 times in 37. Out of 37 people, 18 people cannot do all three. Only 6 can do all three. So 31 can do exactly 2 things.

How do you know that's true though; to me the question read like every other overlapping data set question. It didn't specify 'exactly' two, and seemed to be worded as other questions which mention people being in at least two sets...

Understand here that 37 is the number of people, not the number of instances. Hence 6 is not counted 3 times in 37. Out of 37 people, 18 people cannot do all three. Only 6 can do all three. So 31 can do exactly 2 things.

How do you know that's true though; to me the question read like every other overlapping data set question. It didn't specify 'exactly' two, and seemed to be worded as other questions which mention people being in at least two sets...

The statement given in the question is this: "If the number of people who could do at least two of these things, was found to be 37 and the number of people who could do all these things was found to be 6"

Say, I have 37 people in front of me and I say that these are the people who can do at least two of the three things - say these people are P1, P2, ...P37. I also know that exactly 6 people can do all three things. These 6 are P1, P4, P8, P9, P10, P12 Tell me, how many people can do exactly 2 of the three things? 31 or 19? The answer here is 31.

Note that this situation is different from the usual: 10 people can swim and dance, 20 people can dance and drive and 7 people can swim and drive. In this case, each of the 10, 20 and 7 includes the people who can do all 3 things and hence 10 + 20 + 7 - 6*3 = 19 people can do exactly two things.

But the stem says "at least 2", not exactly. so it should be the formula as follows:

T=n(A)+n(B)+n(C)-n(at least two of the events)+n(All 3 Events)+n(None of the events)

so 144=55+44+53-37+6+X

144=152-37+6+X 144=121+X X=23

OA is incorrect,

Using the formula is a bad idea if you don't understand exactly when and how to use it. If you understand exactly when and how to use the formula, then you would find it too cumbersome to use it and will anyway prefer to reason out the answer!

In your formula, n(at least two of the events) is the sum of the intersection of the circles. This means each intersection includes the area where only two overlap and where all 3 overlap. To check out the two formulas, check out this link: a-school-has-3-classes-math-class-has-14-students-150221.html#p1207266 After you check out the link, note that in your formula, n(at least two of the events) = (d + g) + (e + g) + (f + g) whereas the 37 given to you in this question is (d + e + f + g)