Information from a survey on 175 persons about the goods, services and prices of a supermarket is as follows:

Question

110 persons complained about the prices
67 persons complained about the services
55 persons complained about the goods
20 persons complained about goods and prices only
11 persons complained about services and prices only
16 persons complained about goods and services only
2 persons had no complaints
x persons complained about all three

(i) Draw a Venn Diagram to represent the information from the survey
Calculate the value of x

Answer 1

Although I cannot draw the diagram here, I can give you some information.

Three overlapping circles should be drawn representing the first three categories. The overlap between any two circles represents the three combinations of two complaints, and the intersection of all three would represent x. Another small circle outside these three would represent the 2 with no complaints.

I hope this helps. Thanks for asking.

Answer 2

Draw a large rectangle.
Inside that rectangle 3 intersecting circles. You should show an overlap between any two of the circles and also an overlap of all 3 circles.
Label your circles P,S, and G for prices, services and goods.

Start by placing x inside the overlap of all 3 circles.
Now lets look at the overlap between P and G. That overlap should contain 20, but we have already accounted for x of those, so place 20-x into the open space of the overlap of P and G

Now do the same thing for the overlaps of P and S, and S and G.

Lets go back to circle S, it should have 3 entries so far, x, 11-x, and 16-x

but all of S should contain 67.
So the open area not yet labeled of S would be 67 - x - (11-x) - (16-x)
which simplifies to 40+x
enter that in the un-labeled region of S

Repeat that with the un-labeled regions of P and G

Finally enter 2 into the region of the rectangle not covered by any circle.

Now form an equation by adding up all the components you have entered in your diagram and set it equal to 175, then solve.

........

There is a formula way of doing this, but I am not sure if your studying this topic at that advanced stage.
It would be
N(A U B U C) = N(A) + N(B) + N(C) - N(A ∩ B) - N(A ∩ C) - N(B ∩ C) + N(A ∩ B ∩ C)
175-2 = 110+67+55 - (20-x) - (11-x) - (16-x) + x

You should end up with the same equation as above.

Answer 3

Lisa, I have done your equation twice now and ended up with x=-12
This is course is a nonsense answer.

So either I made an error in my arithmetic or there is a typo in the original question. It could be that the question contains inconsistent data.
Here is my equation

79+x + 19+x + 40+x + 11-x + 16-x + 20-x + x + 2 = 175
x + 187 = 175
x = -12 ??????

Perhaps one of the other tutors could find the flaw.