Notice how many farmers are in each of the 8
regions and what they have and/or have not grown:
"a" farmers have grown vegetables but have not
grown flowers or sugar cane.
"b" farmers have grown Vegetables and flowers but
have not grown sugar cane.
"c" farmers have grown flowers but have not grown
vegetables or sugar cane.
"d" farmers have grown Vegetables and sugarcane
but have not grown flowers.
"x" farmers have grown all three. That's the number
we are wanting to find.
"e" farmers have grown flowers and sugarcane but
have not grown vegetables.
"g" farmers have grown sugarcane but have not
grown vegetables or flowers.
"h" farmers have not grown any of the three things
(perhaps they have only grown fruit and/or cotton).
Let x = the answer, the number who have grown all
three. That's the region right in the center that
is common to all three.
27 farmers have grown vegetables & sugarcane.
d+x = 27
d = 27-x
So we write 27-x for d
24 farmers have grown flowers & sugarcane,
e+x = 24
e = 24-x
So we write 24-x for e
45 farmers have grown vegetables & flowers,
b+x = 45
b = 45-x
So we write 45-x for b
45 have grown sugarcane,
d+x+e+g = 45
Substituting d=27-x, e=24-x,
27-x+x+24-x+g = 45
51-x+g = 45
g = x-6
So we write x-6 for g
63 have grown flowers,
b+c+x+e = 63
Substituting b=45-x, e=24-x,
45-x+c+x+24-x = 63
69-x+c = 63
c = x-6
So we write x-6 for c, also
93 farmers have grown vegetables,
a+b+d+x = 93
Substituting b=45-x, d=27-x,
a+45-x+27-x+x = 93
a+72-x = 93
a = x+21
So we write x+21 for a
...120 farmers
a+b+c+d+x+e+g+h = 120
Substituting a=x+21, b=45-x, c=x-6,
d=27-x, e=24-x, g=x-6,
x+21+45-x+x-6+x+27-x+x+24-x+x-6+h = 120
x+105+h = 120
x+h = 15
So the number of farmers who planted either all of
the three or none of the three totals 15.
Therefore, since you state that the correct answer
is 15, that means that there were no farmers who
planted none of them.
The problem should have stated that, because it is
very feasible that some of the farmers might
not have planted either vegetables, flowers or
sugarcane, but only fruit and/or cotton. Or, they
could have only raised chickens and cows!
So obviously we are assuming h=0. And when, but ONLY
when we do, does x=15.
You should point out to your teacher that it should
have been stated that all farmers planted at least
one of the three. Otherwise we cannot know that h=0.
Edwin
