Lesson SET THEORY REMOVE DUPLICATES AND CREATE VENN DIAGRAM FARMER PROBLEM

Source code of 'SET THEORY REMOVE DUPLICATES AND CREATE VENN DIAGRAM FARMER PROBLEM'

This Lesson (SET THEORY REMOVE DUPLICATES AND CREATE VENN DIAGRAM FARMER PROBLEM) was created by by Theo(13342)

In a village, out of 120 farmers, 93 farmers have grown vegetables, 63 farmers have grown flowers, 45 have grown sugarcane, 45 farmers have grown vegetables and flowers, 24 farmers have grown flowers and sugarcane, 27 farmers have grown vegetables and sugarcane. Find how many farmers have grown vegetables, flowers and sugarcane.

The solution to this problem is:
36 farmers have grown vegetables only.
9 farmers have grown flowers only.
9 farmers have grown sugarcane only.
30 farmers have grown vegetables and flowers only.
12 farmers have grown vegetables and sugarcane only.
9 farmers have grown flowers and sugarcane only.
15 farmers have grown vegetables and flowers and sugarcane.
add these up and you get a total of 120.

the explanation for how this solution was obtained is as follows:

there are 120 farmers in total.
when you add up all the numbers of farmers, there should be 120.
no more and no less.

you have the following numbers to work with.

93 grow vegetables.
this includes the farmers who also grow flowers and or sugarcane.
if you subtract the farmers who grow both vegetables and flowers, and you subtract the farmers who grow both vegetables and sugarcane, you are left with:
93 - 45 - 27 = 21 farmers who grow nothing but vegetables.

63 farmers grow flowers.
this includes the farmers who also grow vegetables and or sugarcane.
if you subtract the farmers who grow both vegetables and flowers, and you subtract the farmers who grow both sugarcane and flowers, you are left with:
63 - 45 - 24 = -6 farmers who grow nothing but flowers.

45 farmers grow sugarcane.
this includes the farmers who also grow vegetables and or flowers.
if you subtract the farmers who grow both sugarcane and vegetables, and you subtract the farmers who grow both sugarcane and flowers, you are left with:
45 - 24 - 27 = -6 farmers who grow nothing but sugarcane.

obviously something is wrong.
you subtracted too many farmers.
you can't have a negative number of farmers.

the reason is that some of the farmers grow all 3 crops.
you need to subtract these from the farmers who grow 2 crops because they are being double counted.

how do you find that out?

the easy way is to ask them.
since you can't do that, then use the following formula that will allow you to figure it out.

the general formula is:

(A OR B OR C) = A + B + C - AB - AC - BC + ABC

that formula has been modified to fit the requirements of this problem.
all that has been changed is the name of the individual sets involved.
these are called categories in this problem.

the specific formula for this problem is:

T = V + F + S - VF - VS - SF + VSF

T is the total number of farmers.
V is the farmers who grow vegetables.
This includes VF and VS and VSF
VF is the farmers who grow vegetables and flowers.
VS is the farmers who grow vegetables and sugarcane.
VSF is the farmers who grow vegetables and flowers and sugarcane.

F is the farmers who grow flowers.
This includes VF and SF and VSF
VF is the farmers who grow vegetables and flowers.
SF is the farmers who grow sugarcane and flowers.
VSF is the farmers who grow vegetables and flowers and sugarcane.

S is the farmers who grow sugarcane.
This includes VS and SF and VSF.
VS is the farmers who grow vegetables and sugarcane.
SF is the farmers who grow sugarcane and flowers.
VSF is the farmers who grow vegetables and flowers and sugarcane.

the formula to follow is, once again:

T = V + F + S - VF - VS - SF + VSF

plug in the numbers that you do know and solve for the numbers that you don't know.

You know that:

T = 120
V = 93
F = 63
S = 45
VF = 45
VS = 27
SF = 24
VSF = ?????

You do not know what VSF is, so you will have to solve for that.

your equation of:
T = V + F + S - VF - VS - SF + VSF
becomes:
120 = 93 + 63 + 45 - 45 - 27 - 24 + VSF
simplify by combining like terms to get:
120 = 105 + VSF
solve for VSF to get:
VSF = 15

there are 15 farmers who plant all three crops (vegetables, flowers, and sugarcane).

Now your numbers will come out correctly.

T = 120
V = 93
F = 63
S = 45
VF = 45
VS = 27
SF = 24
VSF = 15

the formula of:

T = V + F + S - VF - VS - SF + VSF

now becomes:

120 = 93 + 63 + 45 - 45 - 27 - 24 + 15 which becomes:
120 = 120

the number are good based on this formula.

now you want to make your venn diagram.
in order to do that, you have to strip out the double and triple counting.

start with VF and VS and SF.

since VSF is also included in VF and VS and SF, it needs to be subtracted from each of them.

you get:

VFO = VF - VSF = 45 - 15 = 30
VSO = VS - VSF = 27 - 15 = 12
SFO = SF - VSF = 24 - 15 = 9

VFO means VF only.  
it does not include VSF.

VSO means VS only.  
it does not include VSF.

SFO means SF only.  
it does not include VSF.

the numbers you now have to work with are:

T = 120
V = 93
F = 63
S = 45
VFO = VF - VSF = 45 - 15 = 30
VSO = VS - VSF = 27 - 15 = 12
SFO = SF - VSF = 24 - 15 = 9
VSF = 15

now you want to look at V and F and S.

since VFO and VSO and VSF are included in V, you need to subtract them out to get:

VO = V - VFO - VSO - VSF = 93 - 30 - 12 - 15 = 36

VO means V only.  
it does not include VFO and VSO and VSF.   
these have been stripped out.

since VSO and SFO and VSF are included in F, you need to subtract them out to get:

FO = F - VFO - SFO - VSF = 63 - 30 - 9 - 15 = 9

FO means F only.  
it does not include VFO and SFO and VSG.
these have been stripped out.

since VSO and SFO and VSF are included in S, you need to subtract them out to get:

SO = S - VSO - SFO - VSF = 45 - 12 - 9 - 15 = 9

SO means S only.
it does not include VSO and SFO and VSF.
these have been stripped out.

you are now working with:

T = 120
VO = V - VFO - VSO - VSF = 93 - 30 - 12 - 15 = 36
FO = F - VFO - SFO - VSF = 63 - 30 - 9 - 15 = 9
SO = S - VSO - SFO - VSF = 45 - 12 - 9 - 15 = 9
VFO = VF - VSF = 45 - 15 = 30
VSO = VS - VSF = 27 - 15 = 12
SFO = SF - VSF = 24 - 15 = 9
VSF = 15

clean this up to get rid of the clutter and you are left with:

T = 120
VO = 36
FO = 9
SO = 9
VFO = 30
VSO = 12
SFO = 9
VSF = 15

now all the categories do not include any elements from any other categories.

most importantly, the numbers now also add up to the total of all farmers.

you get T = 120
you get VO + FO + SO + VFO + FSO + SFO + VSF equals:
36 + 9 + 9 + 30 + 12 + 9 + 15 equals:
120 which is equal to T.

all the double counting has been removed.

the numbers that you see in this table are the numbers that you will see in the venn diagram that is shown below: