Out of a group of 84 travelers, 50 have visited Estonia

Question

Out of a group of 84 travelers, 50 have visited Estonia, 45 have visited Latvia and 28 have visited Lithuania. If 15 of the travelers have visited all three countries and every traveler has visited at least one, how many have visited exactly two of them?

(A) 9
(B) 12
(C) 24
(D) 39
(E) 45

(A) 9 
(B) 12 
(C) 24 
(D) 39 
(E) 45

Ravz · Accepted Answer

Total travellers = 84
But 15 of them have already visited 3 countries - so we must subtract this number from the following
Total traveller left = 84 - 15 = 69
Estonia = 50 - 15 = 35
Latvia = 45 - 15 - 30
Lithuania = 28 - 15 = 13
Sum (travellers remaining from three countries)= 35+30+13 = 78

Since the remaining have all visited atleast one country, therefore the difference will be the number of people who have visited two countries, which is
Sum (travellers remaining from three countries) - Total travellers left 
=78 - 69
= 9 travellers

Bunuel · Answer

GMATNinja Video Solution:

https://www.youtube.com/watch?v=TRlVgm7RdmE

mbaaspirantss07 · Answer

Is there another way to solve this?

Bunuel · Answer

Checek this:  ADVANCED OVERLAPPING SETS PROBLEMS

Hope it helps.

KarishmaB · Answer

Responding to a pm:

Total - Neither = A + B + C - (Exact 2) - 2*(Exact 3)
84 - 0 = 50 + 45 + 28 - (Exact 2) - 2*15
Exact 2 = 93 - 84 = 9
 Answer (A)

Method 2:  
People 84, Instances = 50 + 45 + 28 = 123
15 people accounted for 45 instances. 
People left 69, Instances = 78
This means 78 - 69 = 9 instances were allocated to people who already had one. So they visited exactly two countries. 
Answer (A)

MagicMeat · Answer

@KarishmaB 
 Total - Neither = A + B + C - (Exact 2) - 2*(Exact 3) 
Why are we multiplying exactly 3*2??

egmat · Answer

Hi MagicMeat,

The key is what the sum  A + B + C = 50 + 45 + 28  actually counts:  country-visits, not people . Each traveler is counted once for  every  country they visited. So how many times does each type of person show up in that total?

- Visited  exactly one  country: counted  1 time . Nothing to fix.
- Visited  exactly two : counted  2 times , so  1 extra . Subtract them once, which is the  - (Exact 2) .
- Visited  all three : counted  3 times , so  2 extra . Subtract them twice, which is the  - 2*(Exact 3) .

The formula exists to make every person count exactly  once , to match the  84  people. A three-country traveler starts at  3  counts, so you strip  2  away to bring them down to  1 . That is the whole reason for the  ×2 .

A check with tiny numbers.  Suppose one traveler visited all three countries and nobody else exists. Then A + B + C = 1 + 1 + 1 =  3 , but there is only  1  person. Fix it:  3  - (Exact 2 =  0 ) -  2 ×(Exact 3 =  1 ) =  1 . The  ×2  is exactly what turns those 3 counts back into 1 real person.

Plugging in:  84 = 123 - (Exact 2) - 30 , giving  Exact 2 = 9 .

Answer: A

KarishmaB · Answer

This is how 3 overlapping sets work: https://anaprep.com/sets-statistics-thr ... ping-sets/ The (exactly 2) region is d + e + f and the (exactly 3) region is k. When we add A + B + C i.e. the full 3 circles, we are adding d, e and f twice each so we subtract them out once and we are adding the k region 3 times so we need to subtract it out twice. That is why we subtract 2 * (Exactly 3)

Out of a group of 84 travelers, 50 have visited Estonia

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Out of a group of 84 travelers, 50 have visited Estonia

Prep Toolkit

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