Sets

[talaangoshtari]

At Lexington High School, everyone takes at least one language- Spanish, French, or Latin- but no one takes all three languages. If 100 students take Spanish, 80 take French, 40 take Latin, and 22 take exactly two languages, how many students are there?

A.198
B.220
C.242
D.264
E.286

[GMATGuruNY]

At Lexington High School, everyone takes at least one language- Spanish, French, or Latin- but no one takes all three languages. If 100 students take Spanish, 80 take French, 40 take Latin, and 22 take exactly two languages, how many students are there?

A.198
B.220
C.242
D.264
E.286

One formula for 3 overlapping groups:

T = A + B + C - (AB + AC + BC) - 2(ABC)

The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in A, everyone in B, and everyone in C:
Those in exactly 2 of the groups (AB+AC+BC) are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups (ABC) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.

In the problem above:
Let total = T.
Spanish = 100.
French = 80.
Latin = 40.
Exactly 2 of the groups = 22.
All 3 groups = 0.

Plugging these values into the formula, we get:
T = 100 + 80 + 40 - 22 - 2*0
T = 198.

The correct answer is A.

Similar problem:
https://www.beatthegmat.com/og-13-178-vi ... 11188.html

[talaangoshtari]

Hi GMATGuruNY,

I read your post, and the way you provide for solving the problem when it is asking for the number of items in exactly one of three sets is so helpful. But what if the data are not represented in percents like this problem?

A college awarded 76 medals in Football, 30 in Basketball and 40 in Tennis. If these medals went to a total of 116 players and only 6 players got medals in all the three sports, how many received medals in exactly one of the three sports ?

A.56
B.42
C.30
D.27
E.18

T = A + B + C - (AB + AC + BC) - 2(ABC)
116 = 76 + 30 + 40 - X - 2(6)
X=18


# received medals in exactly one of the three sports = 116 - 18 - 6 = 92

[GMATGuruNY]

Hi GMATGuruNY,

I read your post, and the way you provide for solving the problem when it is asking for the number of items in exactly one of three sets is so helpful. But what if the data are not represented in percents like this problem?

A college awarded 76 medals in Football, 30 in Basketball and 40 in Tennis. If these medals went to a total of 116 players and only 6 players got medals in all the three sports, how many received medals in exactly one of the three sports ?

A.56
B.42
C.30
D.27
E.18

T = A + B + C - (AB + AC + BC) - 2(ABC)
116 = 76 + 30 + 40 - X - 2(6)
X=18

# received medals in exactly one of the three sports = 116 - 18 - 6 = 92

The formula works for percents or numbers.
Your solution for this problem is perfect.
There seems to be a typo in the question stem.
I believe that it intends to ask for the number of players who received medals in exactly TWO of the three sports.
If so, then the correct answer is E.

[nikhilgmat31]

yes it seems number of players in exactly 2 games seems to be 18.