Lesson SUBSET PROBLEM INVOLVING SURVEY OF FLOWER GARDENERS WITH SOLUTION AND EXPLANATION

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This lesson solves the following problem involving sets and subsets.

PROBLEM

a survey of flower gardeners showed the following:
87 grew roses, 
27 grew tulips,
26 grew orchids,
15 grew both roses, and tulips,
11 grew both orchids and tulips,
19 grew both roses and orchids,
8 grew all three types,
7 grew none of these three.
create a venn diagram to reflect the data above, label diagram clearly. 
use diagram to answer the questions
a) how many grew only orchids?
b) how many grew both roses and orchids, but no tulips?
c)how many grew only roses.
d) how many grew none of these three, or only tulips?
e) how many flower gardeners were surveyed?

SOLUTION

the number of gardeners in total is equal to 103.
61 grew roses only.
9 grew tulips only.
4 grew orchids only.

7 grew roses and tulips only
3 grew tulips and orchids only
11 grew roses and orchids only

8 grew roses and tulips and orchids

7 grew none of the three (roses, tulips, or orchids)

if you add roses only, and roses and tulips only, and roses and orchids only, and roses and tulips and orchids, then you get a total of 61 + 7 + 11 + 8 = 87

if you add tulips only, and roses and tulips only, and tulips and orchids only, and roses and tulips and orchids, then you get 9 + 7 + 3 + 8 = 27

if you add orchids only, and tulips and orchards only, and roses and orchids only, and roses and tulips and orchids, then you get 4 + 3 + 11 + 8 = 26

these number agree with the numbers initially given, so you can assume that the recalculations was done correctly.

the total number of gardeners is 110.
that would be 103 plus 7 for the number of gardeners who didn't plant any of the 3 above (roses, tulips, orchids).

the answer to the rest of the questions are shown below:

a) 
how many grew only orchids?
4 grew only orchids.

b) 
how many grew both roses and orchids, but no tulips?
11 grew roses and orchids, but no tulips.

c)
how many grew only roses.
61 grew only roses.

d) 
how many grew none of these three, or only tulips?
7 grew none of these three and 9 grew only tulips.
this would be a total of 16 that grew none of the three or only tulips.

e) 
how many flower gardeners were surveyed?
a total of 110.

REASON FOR THE SOLUTION

the assumption of the problem is that the number of gardeners that did two of the three, plus the number of gardeners that did all three, were included in the numbers given for each.

to purify the data set, we needed to eliminate the double counting in order to create pure categories.

under this assumption, the procedure used was as follows:

8 gardeners grew all three.
these 8 were subtracted from the number of gardeners that did two out of three.
the gardeners that did roses and tulips only became 15 - 8 = 7.
the gardeners that did tulips and orchids only became 11 - 8 = 3.
the gardeners that did roses and orchids only 19 - 8 = 11.

our purified categories so far included:

gardeners that planted roses and tulips and orchids = 8
gardeners that planted roses and tulips only = 7
gardeners that planted tulips and orchids only = 3
gardeners that planted roses and orchids only = 11

once we eliminated the gardeners that did all three from the gardeners that did two out of three, we then went and subtracted the gardeners that did two out of 3 only and the gardeners that did three out of three from the gardeners that did each type of flower.

the gardeners that did roses only became 87 - 7 - 11 - 8 = 61.
the gardeners that did tulips only became 27 - 7 - 3 - 8 = 9.
the gardeners that did orchids only became 26 - 3 - 11 - 8 = 4.

we now had purified sets to work with.
each set only included elements that belonged in that set.

the venn diagram for this problem is shown below:
<img src = "http://theo.x10hosting.com/problems/venn_diagram_2.jpg" alt = "$$$$$"/ >

SET THEORY

set theory states that the number of elements in the union of 3 sets is found by use of the following equation:

(A union B union C) = A + B + C - (A intersect B) - (A intersect C) - (B intersect C) + (A intersect B intersect C).

This formula assumes that:

The elements in A include elements in A intersect B plus elements in A intersect C plus elements in A intersect B intersect C.
The elements in B include elements in A intersect B plus elements in B intersect C plus elements in A intersect B intersect C.
The elements in C include elements in A intersect C plus elements in B intersect C plus elements in A intersect B intersect C.

It also assumes that:

The elements in A intersect B include elements in A intersect B intersect C.
The elements in A intersect C include elements in A intersect B intersect C.
The elements in B intersect C include elements in A intersect B intersect C.

The formula is structured to avoid the double counting inherent in these inclusions.

if we take your original problem and assign letters to each situation, then we get:

A = 87 grew roses,
B = 27 grew tulips,
C = 26 grew orchids,
D = 7 grew none of these.
(A intersect B) = 15 grew both roses and tulips,
(A intersect C) = 19 grew both roses and orchids,
(B intersect C) = 11 grew both orchids and tulips,
(A intersect B intersect C) = 8 grew all three types,

If we apply the formula, then we get:

(A union B union C) = A + B + C - (A intersect B) - (A intersect C) - (B intersect C) + (A intersect B intersect C)

substituting numbers for each set listed, gets:

(A union B union C) = 87 + 27 + 26 - 15 - 19 - 11 + 8 = 103.

In the above formula:

A = 87
B = 27
C = 26
A intersect B = 15
A intersect C = 19
B intersect C = 11
A intersect B intersect C = 8

add the 7 that didn't grow any of (roses, tulips, orchids), and you get 103 + 7 = 110 total gardeners.

the set theory provide the same total number of gardeners as we solved for above.  
this confirms that we calculated the total number of gardeners correctly.