Out of the 150 students at Rocket Brown Elementary School, 90 joined

Question

Out of the 150 students at Rocket Brown Elementary School, 90 joined the football team, 50 joined the tennis team, 40 joined the hockey team, and 70 joined exactly two of these teams.

Select for  None  the maximum number of students who could have chosen not to join any of the three teams, and select for  All three  the maximum number of students who could have joined all three teams.

Bunuel · Accepted Answer

Official Solution: Check the diagram below: https://gmatclub.com/gmat-focus-tests/resources/import/img/Overlapping%20sets.png The formula for three overlapping sets is: Total = A + B + C - (sum \ of \ overlaps \ between \ exactly \ 2 \ groups) - 2 imes (all \ three) + None . This formula adjusts for overlaps when adding A, B, and C. Sections representing those in exactly two groups (d, e, and f) are counted twice, so we subtract them once to ensure they are only counted once. Similarly, the section representing those in all three groups (g) is counted three times, so we subtract it twice to count it only once. For the original question, substituting the values gives: 150 = 90 + 50 + 40 - 70 - 2 imes (all \ three) + None . 2 imes (all \ three) + 40 = None . Notice that the above implies maximizing All three also maximizes None , and vice versa. Essentially, when one is maximized, both are maximized together. Further notice that the maximum number of students who could have chosen not to join any of the three teams cannot exceed 60, as we know that out of 150 students, 90 joined the football team. But can the maximum for None actually be 60? For this to happen, all students in the hockey and tennis teams would need to also be part of the football team, ensuring that no additional students are outside the football team. If the maximum for None is 60, using the equation 2 imes (all \ three) + 40 = None , it follows that All three would be 10. This gives the following scenario: https://gmatclub.com/gmat-focus-tests/resources/import/img/I06-19.png Thus, this is a perfectly valid case. Since we concluded that maximizing None also maximizes All three , the maximum for All three is 10. Correct answer: None "60" All three "10" GMAT-Club-Forum-kovhbubm.png

VivekSri · Answer

chetan2u · Answer

With 96% incorrect out of first 24 timers taken, the question may seem to be very tough, but can be simplified by Venn Diagram.

Otherwise, the solution would be

Different components:  Only h, Only t, Only f, Only h&t, Only h&f, Only t&f, all three, None

Given: 
Total = Only h + Only t + Only f + Only h&t + Only h&f + Only t&f+ all three + None  = 150
H = Only h + Only h&t + Only h&f + all three = 40
T = Only t + Only h&t + Only t&f + all three = 50
F = Only f + Only f&t + Only h&f + all three = 90
Only h&t + Only h&f + Only t&f = 70

So, just re arranging the total....
H+T+F=40+50+90 = 180
(Only h + Only h&t + Only h&f + all three) + (Only t + Only h&t + Only t&f + all three) + (Only f + Only f&t + Only h&f + all three) = 180
Only h + Only t + Only f + 2(Only h&t + Only h&f + Only t&f) + 3(all three)  = 180
(Only h + Only t + Only f + Only h&t + Only h&f + Only t&f + all three) + (Only h&t + Only h&f + Only t&f) + 2*all three  = 180
(150 - None) + 70 + 2*all three = 180
 None - 2* all three = 40

As we increase None, 'all three will also increase'. Thus, for All  three_{max} , we will get  None_{max} 
As 90 play football and 90 is largest set of people given, the none could be all remaining, that is 150-90 = 60.
So 60 - 2*all three = 40 or all three = 10.

Note: If someone wants to know all the parts.  
Since everything is a part of F, all parts that do not contain F are 0.
Total = Only h + Only t + Only f + Only h&t + Only h&f + Only t&f+ all three + None  = 150
H = Only h + Only h&t + Only h&f + all three = 0 + 0 + Only h&f + 10 = 40........Only h&f = 30
T = Only t + Only h&t + Only t&f + all three =  0 + 0 + Only t&f + 10 = 50........Only t&f = 40
F = Only f + Only f&t + Only h&f + all three =  Only f + 30 + 40 + 10 = 90........Only f = 10

HarshavardhanR · Answer

My approach to this one: https://gmatclub.com/forum/download/file.php?id=82020 We can arrive at N = 2 (III) + 40 Now N cannot be more than 60. 90 students choose football out of 150. So, these 90 cannot be part of NONE. The number of students who choose none of the 3 sports cannot be more than 60. Can it be 60 exactly? For this, we can see what the corresponding values of the other variables are and check that every condition (every equation) is satisfied. When N = 60, III = 10 and I = 10. II is 70 (given). These numbers satisfy every single condition (for example - I + II + III + N = 150). Hence, N can be 60. it is the max possible N value. Answer: Max (all 3 sports) = 10 Max (none of the 3 sports) = 60 ___ Harsha Enthu about all things GMAT | Exploring the GMAT space | My website: gmatanchor.com GMAT-Club-Forum-32ul62wb.png

KarishmaB · Answer

Check out this video on Max-Min in Overlapping Sets first: https://youtu.be/oLKbIyb1ZrI Now note that "All" and "None" both will be maximized together. When "All" is maximum, "None" is maximum too (keeping other overlaps constant). Since of 150, 90 people do play football, None can be at most 150 - 90 = 60 Both other circles will lie inside the football circle in this case. For an overlap of 70 for exactly 2 sports, first let's overlap 10 instances of tennis with 10 of football for an overlap of two sports. Now we have 40 instances each of tennis and hockey and we need another 60 overlap of two sports. Hence 30 of tennis and 30 of hockey should overlap with football independently so that we have a 70 overlap for exactly 2 sports. Now we have 10 instances of both tennis and hockey leftover to overlap with football for the All region. This is what the diagram looks like: Screenshot 2024-04-24 at 6.34.57 PM.png Answer: 10 - Both 60 - None

hemanthPA · Answer

150 = (90+50+40) - (70) -2d + a

150 = 110 -2d+a
-2d = 40 -a

take d = 0, doesnt work
take d = 10, -20 = 40 - 60 = -20 = -20, hence max = 10, none = 60

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Out of the 150 students at Rocket Brown Elementary School, 90 joined

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Two-part Analysis (TPA) Questions

Out of the 150 students at Rocket Brown Elementary School, 90 joined

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards