A survey of 500 farmers showed that of the farmers, 129 grew only wheat, 111 grew only corn, 93 grew only oats, 206 grew wheat, 58 grew wheat and corn, 53 grew wheat and oats, and 178 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) Of the farmers, _ grew at least one of the three crops.

b) Of the farmers,grew all three crops.

c) Of the farmers, did not grow any of the three crops.

d) Of the farmers, ___grew exactly two of the three crops.

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kudzordzifrancis kudzordzifrancis · Answer 1 · 2019-09-08T02:49:11+02:00

Answer:

a) 419

b)34

c)81

d)52

Step-by-step explanation:

Let x represent the number of farmers who grew all three crops and y represent those who grew corn and oath.

This information is represented on a Venn Diagram as shown in the attachment (A)

Let us add all the regions in wheat and equate to 206.

[tex]206 = 129 + 58 - x + x + 53 - x[/tex]

[tex]206 = 240 - x[/tex]

[tex]x = 240 - 206 = 34[/tex]

We can now add all the regions in corns and equate to 178

[tex]58 - x + x + y + 111 = 178[/tex]

[tex]y + 169= 178[/tex]

[tex]y = 9[/tex]

We redraw the Venn diagram as shown in the attachment (B).

We can now answer the question using the second diagram:

Those who grew at least one of the three crops =exactly one +exactly two +exactly three

=121+111+93+24+9+19+34=419

a) Of the farmers, 419 grew at least one of the three crops.

b) Of the farmers,34 grew all three crops.

c) Of the farmers,81 did not grow any of the three crops.

Exactly two=9+19+24=52

d) Of the farmers, 52 grew exactly two of the three crops.