Expert: Bobby Soltani - 8/12/2004

Question
Hello Bobby.
I'd really appreciate if you could explain your approach to solve this question. My problem is that whenever I have a question with too many information like this, I am not sure what I am looking for. It would be great if you could show me the steps to take.
Thank you.

Ang-sui

Question:
In a certain survey of a group of people, 87 people indicated they liked rasberries, 91 indicated they liked strawberries, and 91 indicated they liked blueberries. Of the people surveyed, 9 liked only rasberries, 10 liked only strawberries, 12 liked only blueberries, and 40 liked all three berries. How many of the people surveyed liked both strawberries and blueberries, but not rasberries?

Answer
Hi Ang-Sui,

This one is hard to describe in an email.  It is so much easier to draw a picture.  If you can draw three circles that look like the ones on the left hand side of this page ( http://faculty.vassar.edu/lowry/ch3a.html ).  Instead of x,y,z, label these circles r,b, and s for rasberry, blueberry, and strawberry.  We can use these circles to represent people who like each type of berry.  You can see from the circle that there are seven different categories.  I will show them here as b = blueberry, br = blueberry and rasberry, etc.  The seven categories are:

b
r
s
sb
sr
br
sbr

We are told that 87 people like rasberries, so that means that s + sb + sr + sbr = 87.  Similarly,
b+sb+br+sbr = 91
r+br+sr+sbr = 91
Also, we are given that
b=12
r=9
s=10
sbr = 40
That means that we are only left with three unknowns, sb, br, and sr.  We are trying to find sb to answer the question.
Plugging the knowns into the three equations above gives us
sb+sr = 41
sb+br = 39
sr+br = 38

Solving, let br = 38 - sr
substitute for br in number 2.
sb + 38 - sr = 39
sb - sr = 1
sb = 1 + sr
using equation 1
1 + sr + sr = 41
sr = 20
using equation 1
sb+20 = 41
sb = 21

That is the answer.  Getting it set up is the hardest part.  Draw yourself a picture and it should become clear.  Let me know if you have any questions.

Also, thanks for the great reviews.  I really appreciate them.

Bobby