In the field there are cows, birds, and spiders. Spiders have four eyes and eight legs each. In the field there are 20 eyes and 30 legs. All three animals are presents and there is an odd number of each animal. How many spiders, cows, and birds are presents?

Answer 1

s = numbers of spiders

c = number of cow

b = number of birds

System of equations for eyes:

Spider have four eyes.

Cow have two eyes.

Birds have two eyes.

4 s + 2 c + 2 b = 20

Subtract 2 b to both sides

4 s + 2 c = 20 - 2 b

System of equations for legs:

Spider have eight legs.

Cow have four legs.

Birds have two legs.

8 s + 4 c + 2 b = 30

Divide both sides by 2

4 s + 2 c + b = 15

Subtract b to both sides

4 s + 2 c = 15 - b

4 s + 2 c = 4 s + 2 c

20 - 2 b = 15 - b

20 - 15 = - b + 2 b

5 = b

b = 5

Put this value in equation:

4 s + 2 c = 20 - 2 b

4 s + 2 c = 20 - 2 ∙ 5

4 s + 2 c = 20 - 10

4 s + 2 c = 10

Divide both sides by 2

2 s + c = 5

Since the number of spiders and the number of
cows are odd numbers this will be satisfied only for:

s = 1 and c = 3

If the number of spiders were 3 we would get the equation:

2 s + c = 5

2 ∙ 3 + c = 5

6 + c = 5

c = 5 - 6

c = - 1

It is impossible for the number of cows to be negative.

So:

1 spider , 3 cows and 5 birds