Cows, sheep and ducks

Question

In a field there are cows, sheep and ducks. There are more sheep than ducks. The sheep and ducks between them have a hundred heads and legs (added together), and there are three times as many sheep and ducks added together as there are cows. Find out how many cows, sheep and ducks are in the field

Answer 1

Summer, the number of sheep and ducks added together has to be a multiple of 3, according to the problem.

 

Since each sheep has 5 heads and legs, then we are looking at 5, 10, 15, etc heads and legs.

 

Since each duck has 3, then we are looking at 3, 6, 9, etc.

 

What multiples of 5 and 3 add to 100? We can start with 50 of each and go from there.

 

No, 50 heads and legs of each simply won't work, because we can't have 50 duck heads and legs.

 

We go up by 5 on sheep heads and heads since there are more sheep. 55 sheep heads and legs and 45 duck heads and legs? That makes 11 sheep and 15 ducks, but 11+ 15 = 26, not a multiple of 3.

 

60 sheep and 40 duck. We can't have 40 duck heads and legs.

 

65 sheep and 35 duck? Nope, can't have 35 duck heads and legs.

 

70 sheep and 30 duck? That makes 14 sheep and 10 ducks, and 14 + 10 = 24, which is a multiple of 3!

 

Let's keep going to be sure...

 

75 sheep and 25 duck? Nope.

 

80 sheep and 20 duck? Nope.

 

85 sheep and 15 duck? That makes 17 sheep and 3 ducks, but 17 + 3 = 20, not a multiple of 3.

 

90 sheep and 10 ducks? Nope.

 

95 sheep and 5 duck? Nope yet again.

 

Sooooo, we have 14 sheep, 10 ducks, and 8 cows, since 24 sheep and ducks is three times 8 cows!