I am stuck with the 3 venn diagrams now...
A survey of vegetable gardeners showed the following:
63 grew tomatoes
26 grew cucumbers
38 grew zucchini
18 grew both tomatoes and cucumbers
22 grew both tomatoes and zucchini
19 grew both cucumbers and zucchini
14 grew all three types
18 grew none of these three
(a) How many grew only zucchini?
(b) How many grew both cucumbers and zucchini, but not tomatoes?
(c) How many grew only cucumbers?
(d) How many grew none of these three or only tomatoes?
(e) How many vegetable gardeners were surveyed?
I have found the following aid to work. Cut circles from transparent film, or cello, and label each (tomatores, cukes, etc). For instance, one has a label 63 tomatoes.
Now start to overlap them per the data, labeling each overlap.
It wont take you but a few seconds then to do the math.
To solve this problem, we can use a method known as the Inclusion-Exclusion Principle and draw three overlapping circles to represent the groups of vegetable gardeners.
Let's start by breaking down the given information into three individual categories:
Let's label the circles as:
A: Tomatoes
B: Cucumbers
C: Zucchini
From the information provided, we have the following data:
- 63 grew tomatoes
- 26 grew cucumbers
- 38 grew zucchini
- 18 grew both tomatoes and cucumbers
- 22 grew both tomatoes and zucchini
- 19 grew both cucumbers and zucchini
- 14 grew all three types
- 18 grew none of these three
Now, let's visualize this information in a Venn diagram:
14 (A, B, C)
_____
/ \
A C
\_____ /
B
Using this Venn diagram, we can solve the given questions:
(a) To find the number of people who grew only zucchini, we need to consider only the region labeled C. Looking at the diagram, we can see that there are two regions labeled C - one overlapping with A and B and the other outside of both A and B. Therefore, the number of people who grew only zucchini is 38 - 22 - 19 + 14 = 11.
(b) To find the number of people who grew both cucumbers and zucchini, but not tomatoes, we need to consider the overlapping region of B and C that is not overlap with A. From the diagram, we can see that this region is labeled B ∩ C. Therefore, the number of people who grew both cucumbers and zucchini, but not tomatoes, is 19.
(c) To find the number of people who grew only cucumbers, we need to consider only the region labeled B. From the diagram, we can see that there are two regions labeled B - one overlapping with A and C and the other outside of both A and C. Therefore, the number of people who grew only cucumbers is 26 - 18 - 19 + 14 = 3.
(d) To find the number of people who grew none of these three or only tomatoes, we need to consider the region outside of all three circles labeled (A ∪ B ∪ C)'). This region only includes people who grew only tomatoes and excludes those who grew none of the three. Therefore, the number of people who grew none of these three or only tomatoes is 63 - 14 = 49.
(e) To find the number of vegetable gardeners surveyed, we need to consider the total sum of all the regions in the Venn diagram. The sum of all the regions (including the overlapping and non-overlapping ones) will give us the total number of individuals surveyed.
Using the information given, we can calculate this by summing up the counts of each region:
Total = (A ∪ B ∪ C) = A + B + C - (A ∩ B) - (A ∩ C) - (B ∩ C) + (A ∩ B ∩ C) + (A' ∩ B' ∩ C')
= 63 + 26 + 38 - 18 - 22 - 19 + 14 + 18
Hence, the total number of vegetable gardeners surveyed is 120.