During Lunch, the cafeteria sells 12 sandwiches,
10 soups, and 9 salads
Six students had sandwiches and soup,
4 students had sandwiches and salads,
... 5 students had soup and salad, and
... 2 students had all three. How many
students ate lunch?
We need a Venn diagram. This is the best I can do.
Each rectangle holds the number of students who
bought a particular item. The intersections of
the rectangles (regions) will hold the number
of students who bought the items covered by
multiple rectangles.
Soup
+----------------------+
| a |
| |
| +----------------------+
| | f | |
+------|--------|-------+ | |
| | d | g | | |
| | | | | | Sandwich
| +--------|-------|-------+ |
| | | |
Salad | | e | c |
| b | | |
| +-------|---------------+
| |
+-----------------------+
Let's define each of the regions.
a is the number of students who bought only soup
b is the number of students who bought only a salad
c is the number of students who bought only a sandwich
d is the number of students who bought only soup and a salad
e is the number of students who bought only a salad and a sandwich
f is the number of students who bought only soup and a sandwich
g is the number of students who bought all three items
Now, we show the aggregates of the regions representing
the students who bought various combinations.
cefg = 12 sandwiches (all four of those regions
are contained within the sandwich rectangle)
adfg = 10 bought soup
bdeg = 9 bought a salad
fg = 6 bought soup and a sandwich
ge = 4 bought a salad and a sandwich
dg = 5 bought soup and a salad
g = 2 bought all three items
We were given the number of students in region g, those who
bought all three items. We can insert the number 2 into that
region.
Soup
+----------------------+
| a |
| |
| +----------------------+
| | f | |
+------|--------|-------+ | |
| | d | g | | |
| | | 2 | | | Sandwich
| +--------|-------|-------+ |
| | | |
Salad | | e | c |
| b | | |
| +-------|---------------+
| |
+-----------------------+
(I am restricted to 8000 characters, so I am forced to eliminate
the remaining diagrams. I hope you can follow along
without them.)
By subtracting the smaller regions from larger combinations
of regions, we can begin to determine the number of students
in each region.
Subracting g (2) from dg (5), we find that d represents 3 students who
bought ONLY soup and a salad.
Subracting g (2) from ge (4), we find that e represents 2 students who
bought ONLY a salad and a sandwich.
Subracting g (2) from fg (6), we find that f represents 4 students who
bought ONLY soup and a sandwich.
Subracting d (3), e (2) and g (2) from bdeg (9), we find that b represents
2 students who bought ONLY a salad.
Subracting d (3), f (4) and g (2) from adfg (10), we find that a represents
1 student who bought ONLY soup.
Finally, subracting e (2), f (4) and g (2) from cefg (12), we find that c
represents 4 students who bought ONLY a sandwich.
To find out how many students ate lunch (or at least bought lunch),
we add the numbers from all of the regions in the Venn diagram.
a + b + c + d + e + f + g = 1 + 2 + 4 + 3 + 2 + 4 + 2
a + b + c + d + e + f + g = 18
18 students ate lunch