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Official Guide problem

by Joy Shaha » Sun Jan 22, 2017 1:29 pm
Q. Last year 26 members of a certain club traveled to England, 26 members
traveled to France, and 32 members traveled to Italy. Last year no
members of the club traveled to both England and France, 6 members
traveled to both England and Italy, and 11 members traveled to both
France and Italy. How many members of the club traveled to at least one of
these three countries last year?
A) 52 B) 67 C) 71 D) 73 E) 79
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by DavidG@VeritasPrep » Sun Jan 22, 2017 4:47 pm
Joy Shaha wrote:Q. Last year 26 members of a certain club traveled to England, 26 members
traveled to France, and 32 members traveled to Italy. Last year no
members of the club traveled to both England and France, 6 members
traveled to both England and Italy, and 11 members traveled to both
France and Italy. How many members of the club traveled to at least one of
these three countries last year?
A) 52 B) 67 C) 71 D) 73 E) 79
Let's make a Venn diagram and work from the middle out. If no one visited both England and France, then the England/France/Italy overlap be 0, and the England/France Overlap will be a 0. The initial diagram will look like this:
Image

If 26 members visited England, and 6 went to England and Italy, that leaves 20 who visited England alone
If 26 members visited France, and 11 went to France and Italy, that leaves 15 who visited France alone
Similarly, if 32 members visited Italy, and 6+11 = 17 visited Italy and another country, then 15 visited Italy alone. Now we have the following:


Image


Now add up the regions: 11 + 6 + 15 + 15 + 20 = 67. The answer is B
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by Jay@ManhattanReview » Sun Jan 22, 2017 8:49 pm
Joy Shaha wrote:Q. Last year 26 members of a certain club traveled to England, 26 members
traveled to France, and 32 members traveled to Italy. Last year no
members of the club traveled to both England and France, 6 members
traveled to both England and Italy, and 11 members traveled to both
France and Italy. How many members of the club traveled to at least one of
these three countries last year?
A) 52 B) 67 C) 71 D) 73 E) 79
This is a common set problem.

We know that if three sets A, B, and C intersect, we have:

A OR B OR C = A + B + C - (A & B) - (A & C) - (B & C) + (A & B & C)

Here,

A = Members traveled to England = 26
B = Members traveled to France = 26
C = Members traveled to Italy = 32

A & B = Members traveled to both England and France = 0
A & C = Members traveled to both England and Italy = 6
B & C = Members traveled to both France and Italy = 11

A & B & C = Members traveled to all the three countries = 0 [Since no-one visited both England (A) and France (B), thus A & B & C = 0]

The question asks us to get the number of members who visited at least one of the three countries.

At least of the three countries = Only one country + Only two countries + all the three countries = A OR B OR C

Thus, A OR B OR C = 26 + 26 + 32 - 0 - 6 -11 - 0 = 67.

Answer: B

-Jay
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by DavidG@VeritasPrep » Mon Jan 23, 2017 6:34 am
For another good 3-set overlap question, see here: https://www.beatthegmat.com/a-survey-was ... 70039.html
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by Jeff@TargetTestPrep » Fri Jan 27, 2017 9:51 am
Joy Shaha wrote:Q. Last year 26 members of a certain club traveled to England, 26 members
traveled to France, and 32 members traveled to Italy. Last year no
members of the club traveled to both England and France, 6 members
traveled to both England and Italy, and 11 members traveled to both
France and Italy. How many members of the club traveled to at least one of
these three countries last year?
A) 52 B) 67 C) 71 D) 73 E) 79
We are given the following:

England travelers = 26

France travelers = 26

Italy travelers = 32

England and France travelers = 0

England and Italy travelers = 6

France and Italy travelers = 11

Although it's not stated directly, we can determine that 0 people traveled to all 3 countries because 0 people traveled to both England and France.

To determine how many people traveled to at least one country, we are actually determining the total number of travelers, since each traveler did travel to at least one country. We can do this with the following formula:

Total travelers = England + France + Italy - Sum of (Exactly two countries) - 2 times (All three countries)

Total travelers = 26 + 26 + 32 - (6 + 11 + 0) - 2(0)

Total travelers = 84 - 17 - 0 = 67

Thus, 67 people traveled to at least one country.

Answer: B

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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