a survey of 500 farmers showed that 125 farmers grew only wheat, (cont.)

Question

A survey of 500 farmers showed that 125 farmers grew only wheat, 110 farmers grew only corn and 90 grew only oats. 200 farmers grew wheat, 60 great wheat and corn, 50 grew wheat and oats, 180 grew corn.

How many grew at least one of the three?

How many grew all three?

How many did not grow any of the three?

How many grew exactly two of the three?

How did you get your numbers?

Answer 1

In the Venn diagram shown above the letters mean the number of farmers growing:

A only wheat

B only corn

C only oats

D only wheat and oats

E only corn and oats

F only wheat and corn

G wheat, corn and oats

H none of wheat, corn or oats

Now we put figures to these:

A=125, B=110, C=90;

A+D+F+G=200 (wheat farmers);

F+G=60 (wheat and corn farmers), D+G=50 (wheat and oat farmers);

B+E+F+G=180 (corn farmers);

A+B+C+D+F+G+H=500.

Now we work out more figures.

A+D+F+G=200 and F+G=60, so A+D=140, and since A=125, D=15. D+G=50, so G=35.

B+E=180-60=120, E=120-110=10; F=60-35=25.

Summary so far: (A,B,C,D,E,F,G)=(125,110,90,15,10,25,35). All that remains is H.

A+B+C+D+E+F+G=410, so H=500-410=90.

Q1) 410 grew at least one of the three

Q2) 35 grew all three

Q3) 90 grew none of the three

Q4) D+E+F=15+10+25=50 grew two out of three

Q5) The solution method is explained above (including the diagram)