Farmer Brown had ducks and cows. One day she noticed that the animals had a total of 12 heads and 32 feet. How many of the animals were ducks and how many were

Question

I do not understand this problem. It is not making sense to me.

Answer 1

The first step is usually to assign variables.  Let's call the number of ducks x, and the number of cows y.

 

Both ducks and cows have just a single head apiece, so the total number of heads is just x + y.  There's a total of 12 animal heads about, so we know

 

x + y = 12

 

Ducks, though, have two feet, while cows have 4.  The total number of legs is then 2x (number of feet per duck times number of ducks) + 4y (similarly, number of feet per cow times number of cows), which should come to 32 feet.  So

 

2x + 4y = 32

 

Let's rearrange the first equation to get x = 12 - y.  We can then plug this into the second:

 

2(12 - y) + 4y = 32

24 - 2y + 4y = 32

2y = 8

y = 4

 

Put this back into the (rearranged) first equation:

x = 12 - 4 = 8.

 

This would suggest that there are 8 ducks and 4 cows.

 

Then we check our answer.  Eight ducks have eight heads, and 4 cows have 4, so we have a total of 12 heads, as we expect.  Eight ducks have 16 feet, while four cows have 16 feet, so there's a total of 32 feet, as we expect.  The answer works.