Problem Solving

Survey results:
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 the three foods.

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

any method to solve this? many thanks

There are formulas for this type of overlapping sets problem, one that has three sets. Because of their usefulness, those formulas are some of the few that I bothered to memorize for the GMAT. Memorizing them is pretty easy, because the way the formulas work is pretty obvious.

The best formula in this case, one in which we know how many like two items but in which we don't know how many like exactly two items, is this one.

Total = A + B + C - (The Total Of Those Who Like Two) + (1 x Those Who Like All Three) + (Those Who Like None)

Here's what is going on with that formula. We add the total likes. Then we subtract all the two group overlaps. Then we have to add back the all three number once, because in subtracting the two group overlaps we subtracted the all three overlap three times, thus eliminating it totally, when actually it has to be included one time.

75 = 48 + 45 + 58 - (28 + 37 + 40) + (25) + Those Who Like None

75 = 71 + Those Who Like None

Those Who Like None = 75 - 71 = 4

So now we have those who like none, and we need those who like one.

Pizza: 48 - 28 - 40 + 25 = 5
Hoagies: 45 - 28 - 37 + 25 = 5
Tacos: 58 - 37 - 40 + 25 = 6

4 + 5 + 5 + 6 = 20

Choose D.

A survey was conducted to determine the popularity of 3 food among students. The data collected from 75 students is 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

Draw a VENN DIAGRAM representing the following:
75 Total
48 like pizza
45 like hoagies
58 like tacos
N = ?


Complete the Venn diagram by working from the INSIDE OUT.

25 like all 3 foods:


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:


Subtracting the values in the diagram from P=48, H=45, and T=58, we get:


Subtracting the values in the diagram from Total = 75, we get:


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

The correct answer is D.

BY PICTURE

1-st explain





2-nd explain




3-rd explain






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Another approach:

Anyone who likes exactly one food is counted ONCE.

Anyone who likes exactly two foods is counted TWICE: for instance, if I like pizza and tacos, I'm counted in the pizza group AND the taco group. Since these people are counted twice, they must be subtracted ONCE, since we only want to count each person one time.

Similarly, anyone who likes all three foods is counted THREE times, and must be subtracted TWICE.

Anyone who doesn't like any food isn't counted at all -- they don't appear in any of our groups -- so we have to add them back in.

This means that

(Pizza + Hoagies + Tacos) - (people who like exactly two) - 2*(people who like all three) + (people who like none) = Total

Our only problem here is that we don't have people who are counted exactly twice! The 28 people who like pizza and hoagies INCLUDE the 25 who like all three. So we have to do a little adjusting ...

28 like pizza and hoagies - 25 who like all three = 3 who like only pizza and hoagies
37 like hoagies and tacos - 25 who like all three = 12 who like only hoagies and tacos
40 like pizza and tacos - 25 who like all three = 15 who like only pizza and tacos

So we have

(3 + 12 + 15) who like exactly two. Now our numbers are

(Pizza + Hoagies + Tacos) = 48+45+58
(like exactly two) = 3+12+15
(like all three) = 25
Total = 75

Plugging these into our equation from above, we have

(48 + 45 + 58) - (3 + 12 + 15) - 2*25 + (People who like none) = 75

Which means (People who like none) = 4.

So we have 4 people who like none of the foods, and 75 - 4 - (3 + 12 + 15) - 25, or 16 who like only one.

Hence our answer is 4 + 16, or 20.

Hi everyone,

Here's my video explanation of the problem. Hope you enjoy!

https://youtu.be/fQlnx3SGk04

Rowan