In a class of 80 students, 40% of the students play basketball, 50%

Question

In a class of 80 students, 40% of the students play basketball, 50% play football, and 30% play table tennis. If 5 students play both football and table tennis but not basketball, what is the maximum possible number of students who play both basketball and football, but not table tennis?

A) 11
B) 19
C) 21
D) 24
E) 32

A) 11
B) 19
C) 21
D) 24
E) 32

HarshavardhanR · Answer

I got choice E (32) as answer for this one.

Given that those who play Basketball = 32, those who play Basketball and Football (but not TT) cannot be more than 32. But can this number be 32 itself? Seems like it can.

https://gmatclub.com/forum/download/file.php?id=82531 
___
Harsha

sayan640 · Answer

Thank you HarshR9 , But why did you write minimize a , b , e and g ? Dont think it should be minimize a , e and g only. Thanks a lot for the wonderful solution. GMAT-Club-Forum-wm0wa7wq.png

HarshavardhanR · Answer

Hi sayan640 , My understanding: When we minimize a, e, and g, we are maximizing d. When we are maximizing d (and given minimized g), we are also minimizing b by default. So, in my view , you are correct . One does not have to specifically think about minimizing "b". This should happen by default if we focus on minimizing a, e, and g to maximize d. That said, minimizing "b" is also an objective to be achieved here - that's why I mentioned it. For instance, if b was not at its minimum (say b = 10), then there is no way d would be at its max. So, we need to achieve min (a, e, g, as well as b). Just that minimizing b happens automatically in this case if we focus on the other terms. But you are spot on. Carry on! ___ Harsha Passionate about all things GMAT | Exploring the GMAT space | My website: gmatanchor.com GMAT-Club-Forum-73gses5k.png

sayan640 · Answer

HarshR9 , I have one more doubt . Would you like to point out where I am making mistake here ?

Those who play Basket ball = 32
Those who play football = 40
Those who play T.T = 24
people who play ONLY basketball and football = x
people who play ONLY basketball and TT = y
people who play all three sports = z
80 = 32 + 40 + 24 - ( people who play exactly two sports ) - 2 * ( people who play all three sports)
80 = 96 - ( people who play Basketball and football + people who play football and TT + people who play basketball and TT ) - 2 * ( people who play all three sports)
80 = 96 - ( people who play ONLY Basketball and football + 5 + people who play ONLY basketball and TT ) - 2 * ( people who play all three sports)
80 = 96 - (x +5 +y) - 2*z
x= 96-80-5-y-2*z
x= 11 - (y+2z)

If we want to maximize 'x' , y +2z need to be made minimum i.e zero. Am I making any mistake here ? Kindly point it out. HarshR9 KarishmaB chetan2u
The answer I am getting is = 11 which is incorrect. Why is it incorrect ?

HarshR9

I got choice E (32) as answer for this one.

Given that those who play Basketball = 32, those who play Basketball and Football (but not TT) cannot be more than 32. But can this number be 32 itself? Seems like it can.

___
Harsha
Passionate about all things GMAT | Exploring the GMAT space | My website: gmatanchor.com

Show Spoiler

Attachment:

GMAT-Club-Forum-3avsw7yv.png [ 188.45 KiB | Viewed 2013 times ]

HarshavardhanR · Answer

You arrived at -> 80 = 96 - (x +5 +y) - 2*z Food for thought : You have assumed that there is nobody who does not play a single sport. In reality -> 80 = 96 - (x +5 +y) - 2*z + n (number of students who play none of the sports) So, x = 11 + n - (y+2z). This is why 11 is not the answer here. ___ Harsha Passionate about all things GMAT | Exploring the GMAT space | My website: gmatanchor.com GMAT-Club-Forum-hxh4a6fu.png

sayan640 · Answer

HarshR9  , when you solved it for the first time , how did you come to know all on a sudden , that the no. of players who dont play any sport = 21 ? I would like to know your train of thought while solving this problem .
 chetan2u   KarishmaB  Kindly share some thought in this regard.

HarshavardhanR · Answer

Hi  sayan640

Here is the train of thought I used  for this question . In this specific question, it worked well. For tougher questions, more nuanced logical thought-processes are needed. This question - not so much.

- Given: 32 students play baskeball
- Inference: Number of people who play both basketball and football but not TT (what we are asked to find) cannot be more than 32.
- Observe: All the choices are within 32. 
- Thought: Will 32 (choice E) work? If it does, it has to be the final answer (cannot be more than 32 - is something we have established).
- Try: (refer to my original venn diagram)
If d = 32.
Then, a = e = g = 0.
Also, b = 3.
And, c = 19.

- Now, total = 80 = a + b + c + d + e + f + g + None = 0 + 3 + 19 + 32 + 0 + 5 + 0 + None
- 80 = 59 + None
=> None = 80 - 59 = 21

Number of people playing none of the sports  = 21 is not something we come to know all of a sudden. It follows from the calculation starting with d = 32.

- Observe: With d = 32, everything fits. We have already established than d cannot be more than 32. Thus, 32 must be the final answer here. 
- If this had not worked, I may have tried with the next biggest choice (choice D - 24) or looked at a more nuanced way to solve the question. Here, quick and dirty try with 32 worked.

___
Harsha
Passionate about all things GMAT | Exploring the GMAT space | My website: gmatanchor.com

KarishmaB · Answer

B = 32, F = 40, TT = 24 I want to maximize overlap of F and B so I will simply put B inside F and see if that works. There should be an overlap of 5 between F and TT which can be easily adjusted in the outside 8 of F. So only TT circle would have 24 - 5 = 19 in this case and rest 80 - 40 - 19 = 21 which belong to none. Everything works. Screenshot 2024-07-17 at 12.48.27 PM.png Answer (E)

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.

In a class of 80 students, 40% of the students play basketball, 50%

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 Problem Solving (PS) Questions

­In a class of 80 students, 40% of the students play basketball, 50%

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

In a class of 80 students, 40% of the students play basketball, 50%