BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Set Theory challenge

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
User avatar
Senior | Next Rank: 100 Posts
Posts: 78
Joined: Fri Jul 23, 2010 2:22 am
Thanked: 2 times
Followed by:1 members

Set Theory challenge

by gig92 » Sun Feb 20, 2011 7:57 am
1) In a referendum about three proposals, 78% of the people were against at least one of the proposals. 50% of the people were against proposal n°1, 30% against proposal n°2, and 20% against proposal n°3. If 5% of the people were against all the three proposals, what percentage of people were against more than one of the three proposals?

2) In a club, all the members are free to votre for one, two or three of the candidates. 20% of the members did not vote. 38% of the total members voted for at least two candidates. What percentage of the members voted for either one or three candidates if 10% of the total members who voted, voted for all the three candidates?

3) In a survey shows that 63% of people in a town like cheese whereas 76% like apples. If x% of people like both cheese and apples then find the range of x.

What do you think?
gig92
Top
Source: — Problem Solving |
Master | Next Rank: 500 Posts
Posts: 139
Joined: Fri Nov 26, 2010 4:45 pm
Location: Boston
Thanked: 20 times
Followed by:1 members
GMAT Score:720

by stormier » Sun Feb 20, 2011 8:38 am
gig92 wrote:1) In a referendum about three proposals, 78% of the people were against at least one of the proposals. 50% of the people were against proposal n°1, 30% against proposal n°2, and 20% against proposal n°3. If 5% of the people were against all the three proposals, what percentage of people were against more than one of the three proposals?

What do you think?
78% were against at least one; and
against 1&2&3=5

against 1 + against 2 + against 3 + against 1&2 + against 2&3 + against 1&3 + against 1&2&3 = 78
against 1 + against 2 + against 3 + against 1&2 + against 2&3 + against 1&3 = 73 --- eqn (1)


against 1 + against 1&2 + against 1&3 + against 1&2&3 = 50
against 2 + against 1&2 + against 2&3 + against 1&2&3 = 30
against 3 + against 1&3 + against 2&3 + against 1&2&3 = 20

against 1 + against 1&2 + against 1&3 = 45
against 2 + against 1&2 + against 2&3 = 25
against 3 + against 1&3 + against 2&3 = 15

add the three equations

against 1 + against 2 + against 3 + 2 (against 1&2 + against 1&3 + against 2&3) = 85 ---eqn 2

from eqn 1 and 2

against 1&2 + against 1&3 + against 2&3 = 85 - 73 = 12

% against more than 1 of 3 = against 1&2 + against 1&3 + against 2&3 + against 1&2&3 = 12+5 = 17
Last edited by stormier on Sun Feb 20, 2011 8:39 am, edited 1 time in total.
Top
User avatar
GMAT Instructor
Posts: 1449
Joined: Sat Oct 09, 2010 2:16 pm
Thanked: 59 times
Followed by:33 members

by fskilnik@GMATH » Sun Feb 20, 2011 8:38 am
gig92 wrote:1) In a referendum about three proposals, 78% of the people were against at least one of the proposals. 50% of the people were against proposal n°1, 30% against proposal n°2, and 20% against proposal n°3. If 5% of the people were against all the three proposals, what percentage of people were against more than one of the three proposals?
I suggest you use a very easy-justifiable equation: AuBuC = A+B+C - (Sum of Exactly 2 of them) - 2.(Exactly 3 of them)

Then you will get (Sum of Exactly 2 of them) = 12, therefore the answer will be 12+5 = 17. Justify all that!

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
Top
User avatar
GMAT Instructor
Posts: 1449
Joined: Sat Oct 09, 2010 2:16 pm
Thanked: 59 times
Followed by:33 members

by fskilnik@GMATH » Sun Feb 20, 2011 8:55 am
gig92 wrote:2) In a club, all the members are free to vote for one, two or three of the candidates. 20% of the members did not vote. 38% of the total members voted for at least two candidates. What percentage of the members voted for either one or three candidates if 10% of the total members who voted, voted for all the three candidates?
Hi there!

Let us consider 100 members; focus: (Sum of Exactly 1) + (Exactly 3)

a) Only 80 of them voted.

b) (Sum of Exactly 2) + (Exactly 3) = 38

c) (Exactly 3) = 10% of 80 = 8 members

Therefore:

>> (Sum of Exactly 2) = 38 - 8 = 30

80 = (Sum of Exactly 1) + (Sum of Exactly 2) + (Exactly 3) , then:

>> (Sum of Exactly 1) = 80 - 38 = 42

Answer: 42 + 8 = 50 (%).
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
Top
User avatar
Legendary Member
Posts: 1255
Joined: Fri Nov 07, 2008 2:08 pm
Location: St. Louis
Thanked: 312 times
Followed by:90 members

by Tani » Sun Feb 20, 2011 8:57 am
Let's work with the formula:
G1 + G2 + G3 - G12 - G13 - G23 - 2(G123) +N = T

where G! = # in group 1
G12 = number in both 1 and 2
G123 = number in all three
N = number not in any group
T = total

Where we are dealing only with percents, T = 100

For the first problem, with 78% voting against at least one, we are left with N = 22
G1 = 50, G2 = 30, G3 = 20 and G123 = 5

Plugging in we get

50 + 30 + 20 - G12 - G13 - G23 -2(5) + 22 = 100
simplifying we see G12 + G13 + G23 = 12
Total against more than one = the 12% that voted against 2 proposals plus the 5% that voted against all 3 = 17%.


Problem 2 is similar:

G1 = voted for one; G2 = voted for 2; G3 = voted for 3; N = voted for none.
(there is no G12 because no member can vote for both one and two)

G1 + G2 + G3 + N = 100
N = 20
G2 + G3 = 38 (at LEAST 2 candidates means either 2 or 3)
G3 = 10 therefore G2 = 28
G1 + 28 + 10 + 20 = 100
G1 = 42
Question: how many voted for one or three: G1 + G3 = 42 + 10 = 52


Question 3:

The largest possible number that could like both is limited by the number who like cheese = 63%

To determine the least: G1 (apples) + G2 (cheese) - both + neither = 100%

Assume everyone likes something. (If some liked neither, that would only increase the number who had to like both and we want to find its minimum.)

76 + 63 - both + 0 = 100
Tani Wolff
Top
User avatar
Legendary Member
Posts: 1255
Joined: Fri Nov 07, 2008 2:08 pm
Location: St. Louis
Thanked: 312 times
Followed by:90 members

by Tani » Sun Feb 20, 2011 8:58 am
Let's work with the formula:
G1 + G2 + G3 - G12 - G13 - G23 - 2(G123) +N = T

where G! = # in group 1
G12 = number in both 1 and 2
G123 = number in all three
N = number not in any group
T = total

Where we are dealing only with percents, T = 100

For the first problem, with 78% voting against at least one, we are left with N = 22
G1 = 50, G2 = 30, G3 = 20 and G123 = 5

Plugging in we get

50 + 30 + 20 - G12 - G13 - G23 -2(5) + 22 = 100
simplifying we see G12 + G13 + G23 = 12
Total against more than one = the 12% that voted against 2 proposals plus the 5% that voted against all 3 = 17%.


Problem 2 is similar:

G1 = voted for one; G2 = voted for 2; G3 = voted for 3; N = voted for none.
(there is no G12 because no member can vote for both one and two)

G1 + G2 + G3 + N = 100
N = 20
G2 + G3 = 38 (at LEAST 2 candidates means either 2 or 3)
G3 = 10 therefore G2 = 28
G1 + 28 + 10 + 20 = 100
G1 = 42
Question: how many voted for one or three: G1 + G3 = 42 + 10 = 52


Question 3:

The largest possible number that could like both is limited by the number who like cheese = 63%

To determine the least: G1 (apples) + G2 (cheese) - both + neither = 100%

Assume everyone likes something. (If some liked neither, that would only increase the number who had to like both and we want to find its minimum.)

76 + 63 - both + 0 = 100
Both = 39

Range = 39 to 63

Some prefer to use Venn diagrams for these, but they are impossible to do with typing :-)
Tani Wolff
Top
User avatar
GMAT Instructor
Posts: 1449
Joined: Sat Oct 09, 2010 2:16 pm
Thanked: 59 times
Followed by:33 members

by fskilnik@GMATH » Sun Feb 20, 2011 9:02 am
gig92 wrote:3) In a survey shows that 63% of people in a town like cheese whereas 76% like apples. If x% of people like both cheese and apples then find the range of x.
The maximum value of x occurs when all 63% of the people that like cheese also like apples, therefore xMax = 63%.

From the fact that: AuB = A+B - (A intersection B), then x = 139 - AuB.

We want to find xMin, therefore we need to maximize AuB... take AuB = 100 (in other words, every person likes cheese or apples)!

Therefore xMin = 39%, and the range asked is xMax - xMin = 63% - 39% = 24%
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
Top
User avatar
GMAT Instructor
Posts: 1449
Joined: Sat Oct 09, 2010 2:16 pm
Thanked: 59 times
Followed by:33 members

by fskilnik@GMATH » Sun Feb 20, 2011 9:06 am
Hi, Tani.

In question 2 there was a trap...
Tani Wolff - Kaplan wrote:G3 = 10 therefore G2 = 28
The question stem says 10% of the voters, not of all people involved.

Therefore G3 = 10% of 80 = 8, not 10.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
Top
User avatar
Legendary Member
Posts: 1255
Joined: Fri Nov 07, 2008 2:08 pm
Location: St. Louis
Thanked: 312 times
Followed by:90 members

by Tani » Sun Feb 20, 2011 9:35 am
AHA! A trap - and I fell into it! :-(
Tani Wolff
Top
User avatar
GMAT Instructor
Posts: 1449
Joined: Sat Oct 09, 2010 2:16 pm
Thanked: 59 times
Followed by:33 members

by fskilnik@GMATH » Sun Feb 20, 2011 9:41 am
Tani Wolff - Kaplan wrote:AHA! A trap - and I fell into it! :-(
Yep, but that does not invalidade your good arguments.

See you in other posts!

Cheers,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
Top
User avatar
Senior | Next Rank: 100 Posts
Posts: 78
Joined: Fri Jul 23, 2010 2:22 am
Thanked: 2 times
Followed by:1 members

by gig92 » Sun Feb 20, 2011 10:03 am
fskilnik wrote:
gig92 wrote:3) In a survey shows that 63% of people in a town like cheese whereas 76% like apples. If x% of people like both cheese and apples then find the range of x.
The maximum value of x occurs when all 63% of the people that like cheese also like apples, therefore xMax = 63%.

From the fact that: AuB = A+B - (A intersection B), then x = 139 - AuB.

We want to find xMin, therefore we need to maximize AuB... take AuB = 100 (in other words, every person likes cheese or apples)!

Therefore xMin = 39%, and the range asked is xMax - xMin = 63% - 39% = 24%
The answer is:

[spoiler]0<= x <=39%[/spoiler]

How can we explain it?
gig92
Top
User avatar
GMAT Instructor
Posts: 1449
Joined: Sat Oct 09, 2010 2:16 pm
Thanked: 59 times
Followed by:33 members

by fskilnik@GMATH » Sun Feb 20, 2011 10:14 am
gig92 wrote:The answer is: 0<= x <=39%
How can we explain it?
Range means maximum value - minimum value, therefore we are looking for a NUMBER, not for an interval. The question was ill-posed, just to start with the discussion.

As far as the INTERVAL OF POSSIBLE VALUES for x is concerned,

Take:

x = 43%
only cheese = 20%
only apples = 33%
not cheese not apples = 4%

This is a possible scenario (total = 100%), satisfying all conditions posted in the question stem, therefore x COULD BE over 39%. In other words, I´ve proven to you that your "official answer" is wrong, simple as that.

Another good reason to be sure that the "official answer" you have provided is wrong: if x could be zero percent (as you believe), then (only cheese) + (only apples) = 63 + 76 > 100 (%), what is (also) absurd.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
Top
User avatar
Senior | Next Rank: 100 Posts
Posts: 78
Joined: Fri Jul 23, 2010 2:22 am
Thanked: 2 times
Followed by:1 members

by gig92 » Sun Feb 20, 2011 12:42 pm
fskilnik wrote:
gig92 wrote:The answer is: 0<= x <=39%
How can we explain it?
Range means maximum value - minimum value, therefore we are looking for a NUMBER, not for an interval. The question was ill-posed, just to start with the discussion.

As far as the INTERVAL OF POSSIBLE VALUES for x is concerned,

Take:

x = 43%
only cheese = 20%
only apples = 33%
not cheese not apples = 4%

This is a possible scenario (total = 100%), satisfying all conditions posted in the question stem, therefore x COULD BE over 39%. In other words, I´ve proven to you that your "official answer" is wrong, simple as that.

Another good reason to be sure that the "official answer" you have provided is wrong: if x could be zero percent (as you believe), then (only cheese) + (only apples) = 63 + 76 > 100 (%), what is (also) absurd.

Regards,
Fabio.
Thanks, you're right. One cannot always believe a book :)
gig92
Top
Senior | Next Rank: 100 Posts
Posts: 62
Joined: Mon Aug 02, 2010 3:25 pm
Thanked: 3 times

by Taniuca » Sun Feb 20, 2011 2:44 pm
3) In a survey shows that 63% of people in a town like cheese whereas 76% like apples. If x% of people like both cheese and apples then find the range of x.


c= cheese
a=apples
ca= like cheese & apples

63%= c+ca
76%=a+ca
_________
139 = c+a+2ca (equation 1 -from adding the 2 first statements)

100% = c+a+ca (equation 2- from knowing that the whole of the population likes cheese, apples or both only)

________
39% = ca (equation 1 minus equation 2)
Top
User avatar
GMAT Instructor
Posts: 1449
Joined: Sat Oct 09, 2010 2:16 pm
Thanked: 59 times
Followed by:33 members

by fskilnik@GMATH » Mon Feb 21, 2011 4:03 am
gig92 wrote:Thanks, you're right. One cannot always believe a book :)
My pleasure, gig92.

That´s it! As I tell my students, the "only" problem in "blindly trusting" official answers is to forget that a human (therefore imperfect) being was responsible for solving the problem and (perhaps even another one) was responsible for typing its solution!

Just one detail: the range could also be interpreted as the interval of possible values, therefore I should NOT have bothered on that... in other words, if your book answer were "39% <= x% <= 63%", then we could consider it perfect.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
Top
Post new topic Post Reply