Question

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.

Answer 1

A survey of 500 farmers showed the number of farmers who grew at least one of the three crops, grew all three crops, did not grow any of the three crops, and grew exactly two of the three crops.

Step-by-step explanation:

In order to determine the number of farmers who grew at least one of the three crops, we need to add up the number of farmers who grew each individual crop (wheat, corn, and oats) and subtract the number of farmers who grew all three crops. a) Grew at least one of the three crops:

Number of farmers who grew wheat: 206

Number of farmers who grew corn: 178

Number of farmers who grew oats: 93

Number of farmers who grew all three crops: 58

Total number of farmers who grew at least one of the three crops: (206 + 178 + 93) - 58 = 419

b) Grew all three crops:

Number of farmers who grew all three crops: 58

c) Did not grow any of the three crops:

Number of farmers: 500

Number of farmers who grew at least one of the three crops: 419

Number of farmers who did not grow any of the three crops: 500 - 419 = 81

d) Grew exactly two of the three crops:

Number of farmers who grew wheat and corn: 58

Number of farmers who grew wheat and oats: 53

Total number of farmers who grew exactly two of the three crops: 58 + 53 = 111

Answer 2

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.

`206 = 129 + 58 - x + x + 53 - x`

`206 = 240 - x`

`x = 240 - 206 = 34`

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

`58 - x + x + y + 111 = 178`

`y + 169= 178`

`y = 9`

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.