Old farmer Driscoll had some cows and ducks. One day she counted 20 heads and 56 legs. How many cows and ducks did she have?

Question

I'm not sure how to solve this, could anyone help??

Answer 1

I love this kind of problem.

So first we need to think about what this problem is looking for. We want to know the number of ducks (let's call this D) and the number of cows (let's call this C).

If she counted 20 heads one day, that means that C + D = 20

This is our first equation in our system of equations!

We also know that cows have 4 legs and ducks have 2 legs.

So we can say that the number of legs counted = 56 = 4C + 2D

This is because we know that for each Cow there will be 4 legs and for each duck there will be 2.

So now we have:

C + D =20

4C + 2D = 56

The next step is to pick one equation and make it so C is on one side and D is on the other.

So take the first equation and subtract D from both sides.

Then we have C = 20 - D

This is great because now we can plug in this new equation we have for C into the second equation (the one that is 4C +2D=56).

So we have

4(20-D) + 2D = 56

We can distribute the 4 to the parentheses, to get (20 * 4 )- 4D + 2D = 56

This simplifies to 80 - 4D + 2D = 56.

And it simplifies again to 80- 2D = 56.

subtract 80 from both sides and get -2D = 56 - 80

56 - 80 is -24 so now we have -2D=-24

Divide both sides by -2 and we have D= 12 (this is a positive now!)

D=12 means that there are 12 ducks on the farm.

Now we can plug this in to the first equation (C + D = 20)

C + 12 = 20

subtract 12 from both sides and we have C= 8, which means there are 8 cows on the farm.

So the final answer is 8 cows and 12 ducks.