Each of 90 students participated in at least one of three track tryout

Question

Each of 90 students participated in at least one of three track tryouts: high jump, long jump, and 100-meter dash. If 20 students participated in the high jump tryout, 40 students participated in the long-jump tryout, and 60 students participated in the 100-meter dash tryout, and if 5 students participated in all three tryouts, how many students participated in only two of these tryouts?

(A) 25
(B) 20
(C) 15
(D) 10
(E) 5

Show Answer

Official Answer

B

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(A) 25
(B) 20
(C) 15
(D) 10
(E) 5

KarishmaB · Accepted Answer

A venn diagram can show you exactly what you are supposed to do. Ques4.jpg The high jump circle has 20 students. It includes the pink region (only High jump), d, e and 5 (students who tried all 3) The long jump circle has 40 students. It includes the yellow region (only long jump), d, f and 5 (students who tried all 3) The 100 meter dash circle has 60 students. It includes the green region (only 100 meter), e, f and 5 (students who tried all 3) So when you add 20 + 40 + 60, you get 120 which is 30 more than 90. Why? Because in 20+40+60, you have counted, d, e and f twice and 5 thrice. You need to subtract two times 5 and d+e+f once to get 90. 120 - 2*5 - (d+e+f) = 90 (d+e+f) = 20 This is how you get the formula: Group A + Group B + Group C + Neither - Doubles - 2(Triples) = Total

AtlanticGMAT · Answer

Hi,

So the formula that we can work from is:  Group A + Group B + Group C + Neither - Doubles - 2(Triples) = Total

So 20 + 40 + 60 + 0 - Doubles - 2(5) = 90

110-Doubles = 90

Doubles = 20

The above formula is critical. Although it is simple, it is important to really understand it. Once you do, groups will be much easier.

Happy Studies,

HG.

Narenn · Answer

its B

We are asked how much are in only two tryouts that means what is (H and L only) + (L and 100m only) + (H and 100m only)
Total = H + L + 100m - both + All + None
90 = 20 + 40 + 60 - both + 5 + 0
90 = 125 - both ----------> both = 35

35 = (H and L) + (L and 100m) + (H and 100m)

H and L only = H and L - all ------> H and L - 5
L and 100m only = L and 100m - all ------> L and 100m - 5
H and 100m only = H and 100m - all ------> H and 100m - 5

so (H and L only) + (L and 100m only) + (H and 100m only) = (H and L) + (L and 100m) + (H and 100m) - 15 ---------> 35 - 15 = 20.

Regards,

Abhijit.

Shini1 · Answer

I get your method here..however can you please explain the problem mentioned below as well...

The 38 movies in the video store fall into the following three categories: 10 action, 20 drama, and 18 comedy. However, some movies are classified under more than one category: 5 are both action and drama, 3 are both action and comedy, and 4 are both drama and comedy. How many action-drama-comedies are there?

In this question, we are looking for those movies that are in all 3 categories.

The answer I am getting is -1 which is wrong

38=48-12(i got this 12 by adding 5+3+4)-2x 
therefore, 2=-2x

can you please help me fix this?

KarishmaB · Answer

I am glad you asked this question. This is the reason I tell people to not use formulas in sets questions and to rely on their understanding of venn diagrams. The formula discussed above is used in case you know d, e and f i.e. no of people participating in two and only two events. In this question, the given numbers 5, 3 and 4 are not d, e, and f. They are the number of movies which fall into two groups as well as three groups. 5 movies which are both action and drama include those movies which are comedy as well i.e. which fall in all three categories.

In this case, the formula we use is 
P(A u B u C) = P(A) + P(B) + P(C) – P(A n B) – P(A n C) – P(B n C) + P(A n B n C)
38 = 48 - 12 + x
x = 2

Think about this formula. P(A) includes P(A n B) which includes P(A n B n C). To avoid double counting in case of P(A), P(B) and P(C), you subtract P(A n B), P(A n C) and P(B n C) once but now you have subtracted P(A n B n C) thrice which means you haven't accounted for it at all. Hence, you add it back.

AtlanticGMAT · Answer

Hi,

I'm not a big fan of Venn diagrams and have found that for groups, these formulas,  once you really practice them , are big time savers. In this case, the formula would be:

Singles + Doubles + Triples = Total

24 + 12 + Triples = 38

Triples = 2

This is just a derivative of the other formula. You have to play around with this to get the hang of it.  OG 13 PS #178 is a good one to practice with .

Let me know if you have any questions. Happy studies.

HG.

Shini1 · Answer

I get Karishma's method .. Thanks

@ HerrGrau- how n here do you get your singles from

tried the OG prob too thank u ..and do we have your debrief on the site as well ..your verbal is outstanding ..need to know how for that as well ..

AtlanticGMAT · Answer

Hi,

You can find the singles by subtracting all of the overlaps from the fake total (I call it the fake total because it is the total that counts all of the movies that are in different categories). In this case we are told that there are 12 overlaps. You have to multiply that by 2 because each overlap is in two groups (for instance, there are 5 doubles in action and in drama so those 5 doubles take up 10 slots). So the singles are 48-24 = 24. There is no doubt that the "formula" approach to groups takes practice but it does work very well once you are comfortable with the formulas and how the different groups interact. Maybe a combined approach with Venn Diagrams would work for you.

Let me know if you need more advice on this.

HG.

PS: I am planning on writing a debrief but haven't yet found the time:(

PareshGmat · Answer

Refer diagram below

We require to find value of shaded region = a + b + c

(Values in Blue are calculated)

Setting up the equation

20 + 55 - (b+c) + c + 35 - (a+c) = 90

a+b+c = 20

Answer = B

PareshGmat · Answer

I get your method here..however can you please explain the problem mentioned below as well...

The 38 movies in the video store fall into the following three categories: 10 action, 20 drama, and 18 comedy. However, some movies are classified under more than one category: 5 are both action and drama, 3 are both action and comedy, and 4 are both drama and comedy. How many action-drama-comedies are there?

In this question, we are looking for those movies that are in all 3 categories.

The answer I am getting is -1 which is wrong

38=48-12(i got this 12 by adding 5+3+4)-2x 
therefore, 2=-2x

can you please help me fix this?

Refer diagram below

We require to find x

10 + 11 + x + 4 - x + 11 + x = 38

x = 2

AdmitJA · Answer

Hope this helps! :-D

Capture.PNG

Erjan_S · Answer

Liked your diagram a lot... Just one thing - do we really need to fill all areas in circles? or it is just for illustration? I mean once I have this 5-x and 3-x and 4-x for doubles and X is for triples I can go ahead and resolve it for 10 + 20 + 18 - (5-x) - (3-x) - (4-x) - 2x = 38 without a need to deduct what the remaming area of circile action is = 10- (5-x) - (3-x) - x = 2+x.

Erjan_S · Answer

For practical reasons, I wonder there could be more than a problem with more than 3 populations (diagrams)?

JeffTargetTestPrep · Answer

We can use the following formula:

Total = number of high jump + number of long jump + number of 100-meter dash - number who did two events - 2(number who did all 3 events) + number who did zero events

Since we see that each student participated in at least 1 event, the number who did zero events is zero.

Filling in the rest of the equation, we have:

90 = 20 + 40 + 60 - D - 2(5) + 0

90 = 110 - D

D = 20

Answer: B

CEdward · Answer

Total # of athletes = (HJ + Dash + LJ) - (HJ/LJ only + HJ/Dash only + LJ/Dash only) - 2(All 3) + 0 
90 = (20 + 40 + 60) - x - 10 
x = 20

Answer is B

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Each of 90 students participated in at least one of three track tryout

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