A survey was conducted to find out how many people in a hous

Question

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

A) 17
B) 23
C) 29
D) 35
E) 50

fluke · Accepted Answer

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"

Aj85 · Answer

people who could swim =55

People who clould dance =44

people who could drive = 53

total number of driving,swiming and dancing events = 55 +44 +53= 152

6 people have been triple counted for their expertise, same people extra counted 12 times.

(37 - 6) = 31 people have been double counted for their expertise.

total number of different people who could do these different activities = 152 -(12 +31) =109

so people who could do nothing 144 -109 = 35

Answer D

chawlavinu · Answer

I used to take more time to solve this type of question. Fluke's method is awesome and will help to reduce the time to solve.
Thanks Fluke!!!!!!!!

Spidy001 · Answer

#not swim = 89 => #swim = 144-89 = 55
#not dance = 100 => #dance = 144 - 100 = 44
#not drive = 91 => #drive = 144 - 91 = 53

at least two = sum of two's + sum of three's = 37
sum of three's = 6
=> sum of two's = 31

total = S + D + Dr - (sum of two's ) - 2*(sum of three's) + none

144 = 55 + 44 + 53 -31 - 2*6 + none

=> none = 35

Answer is D.

prakhag · Answer

+1 for D.

Use 3 set formula:

Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Found individual numbers as explained in the earlier posts.

saxenaashi · Answer

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.

KarishmaB · Answer

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.

AccipiterQ · Answer

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,

AccipiterQ · Answer

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...

KarishmaB · Answer

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.

KarishmaB · Answer

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)

JeffTargetTestPrep · Answer

We see that 144 - 89 = 55 residents could swim, 144 - 100 = 44 could dance and 144 - 91 = 53 could drive a car. We can use the formula:

Total = #(swim) + #(dance) + #(drive) - #(exactly two groups) - 2 * #(all three groups) + #(neither)

We need to find #(neither), so let’s denote it by n. We have the numbers for all the other components of the formula except for #(exactly two groups). That is because the number 37 represents #(at least two groups), in other words, 37 represents #(exactly two groups) + #(all three groups). So:

37 = #(exactly two groups) + 6

31 = #(exactly two groups)

Now, we can use the formula:

144 = 55 + 44 + 53 - 31 - 2*6 + n

144 = 109 + n

35 = n

Answer: D

Kinshook · Answer

Given: 
1. A survey was conducted to find out how many people in a housing colony of 144 residents could swim, dance and drive a car. 
2. 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.

Asked: 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?

Please find solution in image.

IMO D

Kinshook · Answer

jamifahad

Given: 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.

Asked: 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?

Attachment:

Screenshot 2024-03-07 at 1.25.44 PM.png [ 45.63 KiB | Viewed 712 times ]

The number of people who could do all these things = 6
The number of people who could do exactly 2 of these things = 37 - 6= 31

A survey was conducted to find out how many people in a housing colony of 144 residents could swim, dance and drive a car.
6 + 31 + A + B + C + D = 144
A + B + C + D = 144 - 37 = 107 --------- (1)

X + Y + Z = 31 --------------(2)

The number of people who could not swim was 89
B + C + Z + D = 89 ---------------(3)

The number of people who could not dance was 100
A + C + Y + D = 100 ----------------(4)

The number of people who could not drive a car was 91
A + B + X + D = 91 ----------------(5)

Adding (3) + (4) + (5)
2(A+B+C) + (X+Y+Z) + 3D = 89 + 100 + 91 = 280
2(107-D) + 31 + 3D = 280
214 + 31 + D = 280
D = 35

IMO D

Kinshook · Answer

Given: 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.

Asked: 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?

Total number of people = 144 
The number of people who could swim = 144 - 89 = 55
The number of people who could dance = 144 - 100 = 44
The number of people who could drive a car = 144 - 91 = 53

The number of people who could all things = 6
The number of people who could do exactly 2 things = 37 - 6 = 31

All = C1 + C2 + C3 - (exactly 2) - 2(all 3) + none
144 = 55 + 44 + 53 - 31 - 2*6 + none
none = 144 - 55 - 44 - 53 + 31 + 12 = 35

The number of people could not do any of these things = 35

IMO D

Wadree · Answer

144 people
Cant swim 89..... Can swim 55
Cant dance 100..... Can dance 44
Cant drive 91..... Can drive 53
Can do at least one = 55+53+44 = 152...... BUUUUT....... 
37 can do at least two.... And 6 can do all.... So 37 - 6 = 31 can do exactly any two......
Now..... when we got total 152 that do at least one..... 6 people that can do all....we included them once in 55 that can swim.....once in 44 that can dance...and again once in 53 that can drive....so we included 6 people thrice instead of once....so we need to subtract the 2 xtra 6s from 152......
Again ..... Some of 31.... Lets say 15 can do exactly driving and dancing....and other 16 can do dancing and swimming....it really doesn't matter which combination we take.....so we included 15 once in driving....and again once in dancing......so we included 15 twice instead of once.... so we need to subtract the 1 xtra 15 from 152......same way we included 16 once in dancing....and again once in swimming.....so we included 16 twice instead of once..... So we need to subtract the 1 xtra 16 from 152......
So....we need to subtract 2×6 ....15 and 16 from 152.... Then 152 - 12 - 15 - 16 = 109 can do something...... 
So..... 144 - 109 = 35 can do FREAKIN NOTHING......

Gmat860sanskar · Answer

Tbh, Best approach here is to find exactly two and then solve it with the know formula, I am attaching my approach just for better understanding of the solution. Hope this helps.

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Attachment:

GMAT-Club-Forum-x8c0q4f7.png [ 222.37 KiB | Viewed 33 times ]

A survey was conducted to find out how many people in a hous

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