```Question 198485
(1) Draw a circle that represents the consumers that are going to buy
a new car. What I do sometimes to avoid confusion is to label the
circle right in the center with a "c"
Don't even worry about numbers at this point.
ahead, some of these are also buying a new car. So, now draw a
circle that intersects the refrigerator circle, and label it in the center
with a small "r" Where the circle intersects the new car circle,
label that intersection with "c x r"
Make sure you leave some room for other circles. Don't worry about numbers.
(3) This is the tricky part. A third circle will represent the
washing machine buyers. Some of these are new car buyers also, and
some are refrigerator buyers, so this new circle will intersect both
the "c" circle and the "r" circle. Label the new circle "w"
Label the intersection with "c", "c x w"
Label the intersection with "r", "r + w"
(4) There is one last bit of information: some of the washing machine
buyers are buying all three things. This is the small intersection of
the three circles. If your picture doesn't have this small intersection,
redraw it. It will be "w + c + r". I just draw an arrow pointing to
it and put this label outside the circles.
(5) If the drawing is right, and you can make sense of it, then you
The labels you have are:
"c" ALL the new car buyers
"c x r" ALL those who will buy both car and refrigerator
"w x c" ALL those who will buy washing machine and car
"w x r" ALL those who will buy washing machine and refrigerator
"w x c x r" those who will buy all three things
The really tricky part is to realize that intersections like
"c x r" also INCLUDE the ones that buy all three.
Try to see the circles two at a time, ignoring the third.
I hope this helps:
(A)  How many were going to buy only a car? That means no washer, no frige
I'm told that 7 will buy "c x r" and 15 will buy "c x w"
{{{33 - 7 - 15 = 11}}}
Tricky part: I an subtracting the "c x r x w = 3" group
they are being subtracted twice, so to subtract them once, I need to add
back {{{3}}} so, {{{33 - 7 - 15 + 3 = 14}}}
14 will buy only a car
-------------------------
(B) Those only buying a washer will not buy car or frige.
I need to subtract from {{{w = 34}}} the intersections "w x c"
and "w x r", so I get {{{34 - 15 - 9 = 10}}}, but remember, by
subtracting these two groups, I am subtracting "w x c x r" twice
so, I'll add back 3 once
{{{34 - 15 - 9 + 3 = 13}}}
13 will buy only a washer
---------------------------
(C) How many were going to buy only a refrigerator?
I need to subtract from "r" the intersections "r x w" and "r x c"
{{{18 - 9 - 7 = 12}}}, again, I am subtracting "c x r x w" twice,
so I'll add it back once. {{{18 - 9 - 7 + 3 = 5}}}
5 will buy only a frige
---------------------------
(D) How many were going to buy a car and a washer but not a refrigerator?
"c x w = 15", but some of those will buy friges. How many? It's
the ones who will buy "c x r x w", so subtract 3
{{{15 - 3 = 12}}}
12 will buy "c x w" but not "r"
----------------------------
(E) How many were going to buy none of these items?
Not an easy question. What I have to do is add up the contents
of the three circles WITHOUT overlaps, that is not counting
anything twice.
I can make use of the questions I've answered.
ONLY "c" plus ONLY "w" plus ONLY "r" = {{{14 + 13 + 5 = 32}}}
OK, but this ignores all the intersections, so I have to add in
"c x w" PLUS "c x r" PLUS "r x w" = {{{7 + 15 + 9 = 31}}}, but
by doing this, I have added "c x r x w" three times, so I need to
subtract it twice. {{{7 + 15 + 9 - 3 - 3 = 25}}}
And {{{32 + 25 = 57}}} so, there are 57 buyers without counting
anyone twice. There were 100 consumers surveyed, so {{{100 - 57 = 43}}}
43 will buy none of the items
----------------------------
ONLY "c" PLUS "r" PLUS ONLY "w" PLUS "c x w" MINUS "c x r x w", then I
should get the result {{{57}}}
ONLY "c" = 14
"r" = 18
ONLY "w" = 13
"c x w" = 15
"c x r x w = 3
{{{14 + 18 + 13 + 15 - 3 = 57}}}
OK

```