A survey was conducted to determine the popularity of 3 food

Question

A survey was conducted to determine the popularity of 3 foods among students. The data collected from 75 students are summarized as below

48 like Pizza
45 like Hoagies
58 like tacos
28 like pizza and hoagies
37 like hoagies and tacos
40 like pizza and tacos
25 like all three food

What is the number of students who like none or only one of the foods ?

A. 4 
B. 16 
C. 17 
D. 20 
E. 23

I got this one right but I spent a lot of time playing with numbers. Can someone please show a faster way.

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=18933 Total = A + B + C - (sum \ of \ 2-group \ overlaps) + (all \ three) + Neither . 75 = 48 + 45 + 58 - (28 + 37 + 40) + 25 + Neither --> Neither = 4. Only Pizza = P - (P and H + P and T - All 3) = 48 - (28 + 40 - 25) = 5; Only Hoagies = H - (P and H + H and T - All 3) = 45 - (28 + 37 - 25) = 5; Only Tacos = T - (P and T + H and T - All 3) = 58 - (40 + 37 - 25) = 6. The number of students who like none or only one of the foods = 4 + (5 + 5 + 6) = 20. Answer: D. For more check ADVANCED OVERLAPPING SETS PROBLEMS: advanced-overlapping-sets-problems-144260.html Hope this helps.

violetsplash · Answer

I have edited the question. The total number of students was missing. After that I could solve it as well. The updated solution is attached.

KarishmaB · Answer

Make a Venn diagram. The toughest of sets questions can be done easily through venn diagrams because you can visualize easily.

Draw three overlapping circles (as shown by Bunuel in his post above). Mark set A as Pizzas, set B as Hoagies and set C as Tacos. 
The part where all three overlap (g in the diagram), mark that as 25.
28 like pizza and hoagies. 25 like all three so 3 like only pizzas and hoagies. Mark d as 3.
37 like hoagies and tacos. 25 like all three so 12 like only hoagies and tacos. Mark f as 12.
40 like pizza and tacos. 25 like all three so 15 like only pizza and tacos. Mark e as 15.

From 75, subtract d, e, f and g. This will give the number of students lying in a, b, c and None. This is exactly what we want.
75 - 3 - 12 - 15 - 25 = 20

nitin6305 · Answer

Bunuel  /  VeritasPrepKarishma , is there a simpler method similar to the table method used for such problems? This appeared in the first 10 questions of my test and I was stumped by the numbers involved.

hanschris5 · Answer

How are we assuming about the neither part to be . I mean when we are subtracting (All three) from each of the (2 like region )

we assume that its part of the bigger a , b and c .

I mean suppose we try solving for Exactly two like using the approach above then we have ,

All 2 like = (28-25)= 3 ; (37-25)=12 ; (40-25)=15 ; Therefore 3+12+15=30 .....But I guess its wrong as I am missing out on the Neither Part.

Now If I consider the approach used by Bruneal then I know that Neither =4 ; so Exactly liking 2 items would be 30-4=26 .

Please correct me if I am wrong , also what approach should I take for Neither.

KarishmaB · Answer

The question asks you the sum of 'None' and 'Only 1' i.e. it asks you 'None + Only 1' (irrespective of how many are in None and how many are in Only 1). You just need the sum.
 
You have that the total number of students = 75

75 = None + Only 1 + Only 2 + All 3
To get 'None + Only 1', all you need to do is subtract 'Only 2' and 'All 3' from 75.

'Only 2' =3 + 12 + 15 = 30 (as correctly calculated by you. It does not include 'None')
'All 3' = 25

So None + Only 1 = 75 - 30 - 25 = 20

anon1111 · Answer

There's a faster way of doing it. 
Let x= total with 3 foods
Total with with 2 foods only => (37-x) + (28-x) + (40-x) = 105-3x= 30
Total with 3 = x = 25

Total 0 and 1 = Total - (Total 2 and Total 3) 
= 75 - 30 - 25
= 20

ElCorazon · Answer

I'd appreciate further clarification on why we use the version of the formula where we add back in the 'all' category? I'm not sure why we are able to assume that the 'all' category also includes numbers from the 'two items' sections? Thanks

Bunuel · Answer

To understand the formulas please check  ADVANCED OVERLAPPING SETS PROBLEMS

Hope this helps.

mcelroytutoring · Answer

When you draw the Venn diagram, I suggest starting with the "triple overlap" number of 25 and working your way out from there. Don't forget to adjust the numbers as you go.

Attached is a visual that should help.

Brego · Answer

ScottTargetTestPrep

Hey Scott,

I was wondering why formula given in TTP formula sheet won't work: Unique members = All A + All B + All C - Groups of two - 2 group of three + Neither.

If I plug in I find that Neither is 71.

I solved all ttp questions and didn't get anything like this

75 = 48 + 45 + 58 - (28 +37 +40) - 2(25) + Neither

I also see this formula being cited:Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither

I haven't seen this formula on TTP either

Thank you

ScottTargetTestPrep · Answer

I understand that some of the quantities in the formula need clarification. For instance, in the formula:

Unique members = All A + All B + All C - Groups of two - 2 group of three + Neither

“Groups of two” refers to the number of members who belong to exactly two of the groups. For instance, in the question it says 28 people like pizza and hoagies; however, the 28 people also include people who like pizza, hoagies AND tacos. For the formula to work, you need to take the number of people who like ONLY pizza and hoagies to be 28 - 25 = 3. Similarly for the remaining groups of two.

If you take the number of unique members to be 75; All A, All B and All C to be 48, 45 and 58 respectively; groups of two to be 30 (because there are 28 - 25 = 3 who like pizza and hoagies only, 37 - 25 = 12 who like pizza and taco only, 40 - 25 = 15 who like pizza and taco only, for a total of 3 + 12 + 15 = 30) and groups of 3 to be 25; you’ll get:

75 = (48 + 45 + 58) - 30 - 2(25) + neither

75 = 151 - 30 - 50 + neither

neither = 4

By the way, if you solve 75 = 48 + 45 + 58 - (28 +37 +40) - 2(25) + Neither; you’ll get neither = 79, not 71.

If you’d rather use the formula Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither, then you need to take 48 + 45 + 58 = 105 for the quantity “sum of 2-group overlaps”.

Posted from my mobile device

GMATGuruNY · Answer

Draw a VENN DIAGRAM representing the following:
 75 Total
48 like pizza
45 like hoagies
58 like tacos
N = ? 
 https://i.postimg.cc/xkb3LpVb/pizza-0.png

Complete the Venn diagram by working from the INSIDE OUT.

25 like all 3 foods: 
 https://i.postimg.cc/5Xbh7v20/pizza-1.png

28 like pizza and hoagies
37 like hoagies and tacos
40 like pizza and tacos 
Subtracting from these figures the 25 who like all 3 foods, we get:
 https://i.postimg.cc/njBTfRdt/pizza-2.png

Subtracting the values in the diagram from P=48, H=45, and T=58, we get:
 https://i.postimg.cc/F7RW7Zz0/pizza-3.png

Subtracting the values in the diagram from Total = 75, we get:
 https://i.postimg.cc/DSGcXYT4/pizza-4.png

Thus:
Only P + Only H + Only T + N = 5 + 5 + 6 + 4 = 20.

D

ryjames · Answer

Has this been addressed in the TTP course? I too came across this type of questions and never recalled seeing that equation in TTP.

What is the different between "Unique members" vs "Total"?

ScottTargetTestPrep · Answer

Response:

In the formulas, “Unique Members” and “Total” refer to the same quantity. The former emphasises that adding the number of elements in groups A, B, and C does not necessarily give us the total number of elements, as the elements in overlaps will be counted multiple times.

In the TTP course, we teach the following two formulas:

Total # of Unique Elements = # in Group A only + # in Group B only + # in Group C only - # in Double Overlaps Only + # in Triple Overlap + # in Neither

Total # of Unique Elements = # in (Group A) + # in (Group B) + # in (Group C) - # in (Groups of Exactly Two) - 2  + # in (Neither)

There is also the formula Total # of Unique Elements = A + B + C − (sum of 2−group overlaps) + (all three) + Neither. The difference between this formula and the above formulas is that you can add the number of elements that belong to two or three groups directly, whereas in the above formulas you first have to subtract the number of elements of “Group of Exactly Three” each time before adding them. Using this formula might save you a few seconds, but we don’t think memorizing a third formula is worth the gain.

Vubar · Answer

GMATNinja  Just wanted to check, how would you feel about me sharing the method you (Harry) taught during the 24-hour stream, for tackling this question?

GMATNinja · Answer

Thank you so much for asking, Vubar ! Since we shared it publicly in a video, it's absolutely no problem to post it here, especially if it includes a shoutout to Harry.

And thank you for sticking with the whole marathon, Vubar ! It meant a lot to me.

Vubar · Answer

So it turns out this question doesn't need the whole method that Harry from GMAT Ninja taught (the better Harry

), so if you want to see the whole method, you can find it here: https://youtu.be/FJa6Dz5p1Ko?t=23573 Using the table he suggests, this question becomes very simple. Absolutely watch the video linked above (and if you can, please do support the cause they were fundraising for). To sum up their method, I'd break it down into a few steps: 1a . Identify how many groups you have , with memberships that are overlapping (e.g., in this case, the number of foods liked by any individual). 1b . Label a column "# groups" with the word groups replaced with whatever title is appropriate. 1c . Fill in the numbers for the groups with the information you know from the question. 2a . Make a second column for the number of members (e.g., in this case, the # of students) 2b . Fill in this column with the information you have from the question. Add a total row for the total number of members and fill it in if we know it. 2c . If the question is asking for one of these numbers, mark it with "X" , as this will be the number we build the table around to solve for. In this question, you don't need step 3, but I'll include it for completeness. 3a . Make a third column for the number of memberships including overlaps (e.g., in this case, every food that is liked by every individual). Note: This number will be greater than the total number of people because by definition some people are members of more than one group (e.g., like more than one food). 3b . Fill in this column with any information we know from the question. If the question is asking for one of these numbers, mark it with "X" , as this will be the number we build the table around to solve for. Note: We may need to sum the total memberships to find the value for the total row here (e.g., 48 like Pizza + 45 like Hoagies + 58 like tacos). 4 . Any empty spaces, fill them in by creating expressions using the numbers around them and X. Solve for X . So my summary of Steps 1, 2, and 4 to solve this is in this table: https://cdn.discordapp.com/attachments/954678446494191617/955014427911028746/SPOILER_unknown.png So the answer is D , as there are 20 people who like either 0 or 1 food. Just FYI, if the question required it, possible values for Column 3 (Overlaps) could have been: (X * 0) + Y 60 = 2 foods * 30 members 75 = 3 foods * 25 members --- 151 Thank you Charles! And thank you again, and everyone who helped organise, for making it happen. Trying to help spread the word was the least I could do. I did delay my exam date though, trying to recover the sleep - I am sure you will understand better than most!

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A survey was conducted to determine the popularity of 3 food

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