There are 150 students at Seward High School. 66 students

Question

There are 150 students at Seward High School. 66 students play baseball, 45 play basketball, and 42 play soccer. 27 students play exactly two sports, and three students play all three of the sports. How many of the 150 students play none of the three sports?
A) 0
B) 27
C) 30
D) 99
E) 78

Please help to understand the approach to tackle the venn diagram problems !!!

A. 0
B. 27
C. 30
D. 99
E. 78

Bunuel · Accepted Answer

150 = {baseball} + {basketball} + {soccer} - {exactly 2 sports} - 2*{exactly 3 sports} + {none of the ports} : 150 = 66 + 45 + 42 - 27 - 2*3 + {none of the ports} --> {none of the ports}=30 Answer: C. Look at the diagram: untitled.PNG When we sum {baseball} + {basketball} + {soccer} we count students who play exactly 2 sports (yellow section) twice , so to get rid of double counting we are subtracting {exactly 2 sports} once. Similarly, when we sum {baseball} + {basketball} + {soccer} we count students who play exactly 3 sports (blue section) thrice (as it is the portion of all three groups), so to count this group only once we are subtracting 2*{exactly 3 sports} . For more on this check: https://gmatclub.com/forum/formulae-for- ... ml#p729340 Hope it helps.

gtr022001 · Answer

great explanation using venn diagram +1

prashantbacchewar · Answer

Bunuel can we expect 3 overlapping set question on GMAT and are we supposed to know the formula.

Bunuel · Answer

I've seen several GMAT questions on 3 overlapping sets, so you should understand the concept behind such kind of problems. Check this for more: formulae-for-3-overlapping-sets-69014.html#p729340

AMsac123 · Answer

Hi you try doing question 203 in OG 17 the same way? there the 2*aandband c is not working  Bunuel

Bunuel · Answer

Please check that question  HERE .

Luckisnoexcuse · Answer

150 = 66+45+42 - 27 - 6 + x
x= 30

C

AMsac123 · Answer

Hi  Bunuel  thanks a lot can question 132 be tackled the same way?

Bunuel · Answer

Check  HERE .

All other OG questions are  HERE .

ScottTargetTestPrep · Answer

We can create the following equation:

Total students = # who play baseball + # who play basketball + # who play soccer - # who play exactly two - 2(# who play all 3) + # who play neither

150 = 66 + 45 + 42 - 27 - 2(3) + n

150 = 120 + n

n = 30

Answer: C

0Lucky0 · Answer

Bunuel , I am still unable to understand how 27{exactly 2 sports} is being counted twice when we add Baseball + basketball + soccer ? Even the venn Diagram has 3 yellow parts. So I am confused regarding it. Can you please shed some more light on this one part? I understand the rest of the logic. Thanks

IanStewart , KarishmaB , chetan2u , MartyTargetTestPrep , If possible. Thanks

IanStewart · Answer

There's a better way to do the question, as I'll explain below, but if we have 66 baseball players, 45 basketball players, and 42 soccer players, we have 66+45+42 = 153 players in total, but we've counted some of these players twice. We have 3 who play every sport, so we've counted those people three times. We only want to count them once, so we should subtract 2*3 = 6 from our total. We also have 27 who play exactly two sports, so we've also counted those people twice. We only want to count them once, so we should subtract a further 27 from our total. So we really have only 153 - 6 - 27 = 120 unique players who play one or more of the sports, and since there are 150 students in total, 30 play none of the sports.

But you don't need to think through this problem that way. We can draw a 3-circle Venn diagram:

• we know we have 3 players in the middle, where all the circles overlap
• we know we have 27 players somewhere where pairs of circles overlap. We don't know where these players go, but it  cannot matter  where we put them; if it mattered, i.e. if the answer to the question changed depending where these 27 people are in the Venn diagram, the question would have more than one right answer, and that can never be true of a GMAT PS question. So we can put these 27 anywhere we like, knowing we'll get the right answer no matter what we do. We can put them, say, where baseball and basketball overlap, and then we'll have 0 people where baseball and soccer overlap, and 0 where basketball and soccer overlap
• then it's easy to fill in the rest of the Venn diagram -- we have accounted for 30 of the baseball and basketball players, so 36 play only baseball, and 15 only basketball. We have accounted for only 3 soccer players, so 39 play only soccer.
• now we have an entire Venn diagram filled-in, and adding the numbers in each region, we have 36 + 15 + 39 + 27 + 0 + 0 + 3 = 120 people in the diagram, so 30 must be outside the diagram.

KarishmaB · Answer

Check out this blog post first: https://anaprep.com/sets-statistics-thr ... ping-sets/ "counted twice" means that the elements in this region belong to two sets. In my diagram, the elements in the green region d belong to both Set A and Set B. Similarly, elements in the region e belong to both Set B and Set C. Elements in the region f belong to both Set A and Set C. That is why elements in regions d, e and f are counted twice, because they are counted in both sets to which they belong.

0Lucky0 · Answer

Thanks, It makes sense now.

btsaami · Answer

Method 1: Let x be no. of people playing 1 sport, y-->2 sports and z--> 3 sports.
x+y+z+n=150
x+2y+3z=153 (can be visualized using Venn diagram)

Subtracting the 2 eqns, y+2z-n=3
y=27, z=3, substitute these values we get

n=27+6-3= 30. Answer.

Method 2:  Use distribution logic
150 students, 66 play baseball, 45 basketball and 42 soccer. That totals 66+45+42=153
Now, out of the total 153, we know that 27 students play two sports and 3 students play 3 sports.
Therefore, we can first find no. of students who play only one sport by subtracting 153-2*27-3*3=153-63=90
These 90 students play only one sport, 27 play two sports and 3 play all 3. Subtracting these no.s from 150, we get the total no. of students who play no sports 150-90-27-3= 150-120=30.

Method 3: 
You can draw or just visualize the 3-Venn diagram, each of the 7 element of overlap will add up to total 150.
We can remove reduntant values.

150= 66+45+42-27-2*3+n
n=150-153+27+6
n=30. Hence, the answer.

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There are 150 students at Seward High School. 66 students

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