Asurvey of 500 farmers showed that of the farmers, 125 grew only wheat, 111 grew only corn, 92 grew only oats, 196 grew wheat, 58 grew wheat and corn, 44 grew wheat and oats, and 183 grew corn. determine the number of farmers who a) grew at least one of the three, b) grew all three, c) did not grow any of the three, and d) grew exactly two of the three.

a) 413 farmers

b) 31 farmers

c) 82 farmers

d) 54 farmers

Step-by-step explanation:

Let's call:

a: the number of farmers that only grew wheat, that is equal to 125

b: the number of farmers that only grew corn, that is equal to 111

c: the number of farmers that only grew oats, that is equal to 92

d: The number of farmers that grew wheat and corn but not oats

e: The number of farmers that grew wheat and oats but not corn

f: The number of farmers that grew corn and oats but not wheat

g: The number of farmers that grew wheat, corn and goals

h: the number of farmers that didn't grow any of the three

Adicionaly from the question we know that:

1. 196 grew wheat, so our first equation is:  a+d+e+g=196

2. 58 grew wheat and corn, so our 2nd equation is: d+g=58

3. 44 grew wheat and oats, so our 3rd equation is: e+g=44

4. 183 grew corn, so our 4th equation is: b+d+f+g=183

5. there are 500 farmers, so our 5th equation is: a+b+c+d+e+f+g+h=500

Then replacing a and d+g on the 1st equation we get:

125+58+e=196

e=13

Using this result on the third equation we get:

13+g=44

g=31

with these result we replace on second equation and obtain:

d+31=58

d=27

Then replacing b, d and g on 4th equation we obtain:

111+27+f+31=183

f=14

Finally replacing on the 5th equation we obtain:

125+111+92+27+13+14+31+h=500

h=87

So the values of every event is: a is 125, b is 111, c is 92, d is 27, e is 13, f is 14, g is 31 and h is 87.

The number of farmers who grew at least one of the three are the number of farmers that grew one, two or the three. That's can be calculate as:

a+b+c+d+e+f+g=125+111+92+27+13+14+31=413

The number of farmers who grew all of three are the number of farmers that belong to event g, that means 31 farmers.

The number of farmers that didn't grow any of the three are the number of farmers that belong to event h, that means 87 farmers.

The number of farmers that grew exactly two of the three are the number of farmers that belong to event d, e and f, that means 54 farmers.

d+e+f=27+13+14=54

step-by-step explanation:

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