# Thread: Simple Word Problem Sets, Venn

1. ## Simple Word Problem Sets, Venn

I'm sure this is a super easy problem, but I've been out of school for five years and didn't take much math my first time since I received a B.A. So I just need a little help getting a jumpstart back into it. I've worked this problem over and over just can't get it to go. Thanks for your help.

Assume that 170 surveys are completed. Of those surveyed, 88 responded positively to effectiveness, 93 responded positively to side effects, and 80 responded positively to cost. Also, 47 responded positively to both effectiveness and side effects, 37 to effectiveness and cost, 47 to side effects and cost, and 20 to none of the items. How many responded to all three?

2. Originally Posted by Tainted1
I'm sure this is a super easy problem, but I've been out of school for five years and didn't take much math my first time since I received a B.A. So I just need a little help getting a jumpstart back into it. I've worked this problem over and over just can't get it to go. Thanks for your help.

Assume that 170 surveys are completed. Of those surveyed, 88 responded positively to effectiveness, 93 responded positively to side effects, and 80 responded positively to cost. Also, 47 responded positively to both effectiveness and side effects, 37 to effectiveness and cost, 47 to side effects and cost, and 20 to none of the items. How many responded to all three?

Of these:

150-88-93-80+47+37+47=20

responded to all three (this is an example of the inclusion/exclusion principle)

We start with the number of surveys with response subtract the numbers
of repsonses to each question, but now we have removed those that
responded to more than one question multiple times - twice for those that
answered exactly two questions and three times for those that responded
to all three.

So adding back in the numbers that responded to each pair of questions
we have removed those who answered exactly two questions, and left
one count in for each person who responded to all three.

RonL

3. Here's the diagram.

170-88-93-80+47+47+37-20=20

4. Hello, Tainted1!

No, this is not an easy problem . . . even with Venn diagrams.

170 surveys are completed. .Of those surveyed, their positive responses were:

88 to Effectiveness
93 to Side effects
80 to Cost

47 to Effectiveness and Side effects
37 to Effectiveness and Cost
47 to Side effects and Cost

20 to none of the items.

How many responded to all three?
There is a formula for the number of elements in the union of three sets.

$n(E \cup S \cup C) \;=\;n(E) + n(S) + n(C) - n(E\cap S) - n(S\cap C)$ $- n(E\cap C) + n(E \cap S \cap C)$

Of the 170 surveys, 20 responded to none of the three.
. . Then there are $150$ who responded to at least one.

The formula becomes: . $150 \:=\:88 + 93 + 80 - 47 - 37 - 47 + n(E \cap S \cap C)$

Therefore: . $n(E \cap S \cap C) \:=\:20$

5. ## Thank You

Thank you everyone. It is much clearer now. Math isn't my best subject so I'm sure I'll be back.

Thanks again.

T