Let R = the set of all gardners who grew roses, regardless
of whether they grew the other two flowers.
Let T = the set of all gardners who grew tulips, regardless
of whether they grew the other two flowers.
Let O = the set of all gardners who grew orchids, regardless
of whether they grew the other two flowers.
The most inclusive clue we have is this one:
4 grew all three types
Those 4 are in all three circles, and the only
region that is common to all three circles is
the middle region e, so we put 4 for "e"
Now we look at this:
7 grew both roses and tulips
4 of these 7 are in the middle region,
so the other 7-4 or 3 are in "b", so
we put 3 in region "b":
Next we look at this clue:
5 grew both orchids and tulips
4 of these 5 are in the middle region,
so the other 5-4 or 1 is in "f", so
we put 1 in region "f":
Next we look at this clue:
10 grew both roses and orchids
4 of these 10 are in the middle region,
so the other 10-4 or 6 are in "d", so
we put 6 in region "d":
Next we look at this clue:
45 grew roses
3 of the regions of circle R contain 3,4,
and 6 gardenerss, that's 3+4+6=13 already accounted for,
so the remaining region "a" of circle R must
contain 45-13 = 32. So we put 32 in region "a":
Next we look at this clue:
16 grew tulips
3 of the regions of circle T contain 3,4,
and 1 gardeners, that's 3+4+1=8 already accounted for,
so the remaining region "c" of circle T must
contain 16-8 = 8. So we put 8 in region "c":
Next we look at this clue:
13 grew orchids
3 of the regions of circle O contain 6,4,
and 1 gardeners, that's 6+4+1=11 already accounted for,
so the remaining region "g" of circle O must
contain 13-11 = 2. So we put 2 in region "g":
Next we look at this clue:
9 grew none of these three
That is the region that is outside all three
circles, which is region "h". So we put 9 in
region "h":
Now we have the numbers of gardners in all
8 regions of the Venn diagram identified. So
we can now answer any questions about it:
(a) How many grew only roses?
These are the 32 gardners in the upper left region of R,
because those 32 are not in either of the other two circles.
Answer: 32
(b) How many grew both roses and tulips, but not orchids?
There are 3+4 or 7 gardeners that are in the two parts
where circle R and circle T overlap.
Answer: 7
(c) How many grew only tulips?
These are the 8 gardners in the upper right region of T,
because those 8 are not in any of the other two circles.
Answer: 8
(d) How many grew none of these three or only orchids?
There are 9 who grew none of these three flowers and
there are 2 gardners in the bottom region of O,
because those 2 are not in either of the other two
circles.
Answer: 9+2 = 11
(e) How many flower gardeners were surveyed?
We add up all the gardeners in all eight regions:
32+3+8+6+4+1+2+9 = 65
Answer: 65
Edwin