Question: 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
Correct Answer: B
Solution and Explanation
Approach Solution 1:
As the question says that last year 26 members of a certain club traveled to England. 26 members traveled to France, 32 members traveled to Italy, and 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. In the question, we need to find the number of members of the club who traveled to at least once of all these three countries last year.
So this sum is a sample of a 3-group Overlapping Sets question. It is a question so to solve it, we either can make 3-circle Venn diagram or use the following formula:
Total = (Group 1) + (Group 2) + (Group 3) - (those in 1 and 2) - (those in 1 and 3) - (those in 2 and 3) - 2(those in all 3 groups).
Based on the data, the equation would look like this:
Total = (26) + (26) + (32) - (0) - (6) - (11) - 2(0)
Total = 84 - 17
Total = 67
Approach Solution 2:
n(E or F or I) = n(E) + n(F) + n(I) - n(E and F) - n(F and I) - n(I and E) +n(E and F and I)
n(E or F or I) = 26 + 26 + 32 - 0 - 11 - 6 + 0
Why is the number of people who traveled to E and F and I 0? Because no one traveled to both England and France. So obviously, no one would have traveled to England, France and Italy.
n(E or F or I) = 67
Approach Solution 3:
a+b= 6
b+c = 11
Refer England team: a+b+ (persons who only went to England) = 26
Persons who only went to England = 26 - (a+b) = 26-6 = 20
Refer French team: b+c+(persons who only went to France) = 26
Persons who only went to France = 26 - (b+c) = 15
Refer Italian team: a+b+c+f = 32
Now with above info.
Members of the club traveled to at least one of these three countries = Nothing but summation of all values (take only once)
Leftmost circle (England) : (20+a+b)
French: 15+c (all remaining already covered in previous circle)
Italy: f (all other variables taken in previous two circles)
(20+a+b)+(15+c)+f will give the answer
20+15+ a+b+c+f = 35+32 = 67
“Last year 26 members of a certain club traveled to England, 26 members”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide". To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.
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