100 people seated at different tables in a mexican restauant were asked if their party had ordered any of the following items: margaritas, chili con quesadillas

23 people had ordered none of these items.

29 people had ordered chili con queso or quesadillas but did not order margaritas.

41 people had ordered quesadillas.

46 people had ordered at leasdt two of these items.

13 people had ordered margaritas and quesadillas but had not odered chili con queso

26  people had ordered margaritas  and chili con queso.

Expert Answers

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There is not enough information to completely solve the problem as given.

Let M represent customers who ordered margaritas, Q customers who ordered quesadillas, and Ch customers who ordered chili.

Let a be people only in M, b people in `M nn Ch` , c people who ordered all 3, d people in Ch only, e people in `Ch nn Q` , f people in Q only, and g people in `M nn Q` .

Draw a Venn diagram with 3 circles labeled M,Q, and Ch.

We have the following:


g=13 (13 people ordered M and Q, but not Ch.)

c+e+f+g=41 => c+e+f=28 (Q has 41 people total.)

d+e+f=29 (29 people ordered Ch and Q but not M)

b+c=26 (26 people ordered Ch and M)

b+c+e+g=46 => b+c+e=33 (46 people ordered at least 2)

Since b+c=26 e=7.

Now a+b+c+d+e+f+g=77; we know g=13 and e=7 so
a+b+c+d+7+f+13=77 => a+b+c+d+f=57
But b+c=26 and d+f=22 (d+e+f=29 and e=7) so a=9.

So far we have a=9,e=7,g=13.

Our remaining equations are:

b+c=26 which is 3 equations and 4 unknowns. There cannot be a unique solution for these equations.

Suppose c=13; then b=13, d=14, and f=8. This solution satisfies all constraints:

c+e+f+g=13+7+8+13=41 so 41 ordered Q.

d+e+f=29 so 29 ordered Ch and Q.

b+c=26 so 26 ordered Ch and M .

g=13 so 13 ordered M and Q but not Ch

b+c+e+g=13+13+7+13=46 so 46 ordered at least 2

a+b+c+d+e+f+g=9+13+13+14+7+8+13=77 so 23 ordered none.


Another solution would be a=9,b=14,c=12,d=13,e=7,f=9,g=13

You will see that the constraints are again satisfied.

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