Question

A farmer notes that in a field full of horses and geese there are 30 heads and 84 feet. How many of each animal are there?

Answer 1

There are 12 horses and 18 geese.

Explanation:

We are given that in a field full of horses and geese, a farmer notes that there are 30 heads and 84 feet.

We can write a system of equations using the given information.

Let the amount of horses there are be represented by h and geese by g.

Assuming each horse and geese has only one head, we can write that:

`\displaystyle h + g= 30`

And assuming that each horse has four feet and each geese has two feet, we can write that:

`4h + 2g = 84`

This yields a system of equations:

`\displaystyle \left\{\begin{array}{l} h + g = 30 \\ 4h + 2g = 84 \end{array}`

We can solve it using substitution. From the first equation, isolate either variable:

`g = 30 - h`

From the second, we can first divide by two:

`2h + g = 42`

And substitute:

`2h + (30 - h) = 42`

Combine like terms:

`h + 30 = 42`

Subtract:

`h = 12`

Therefore, there are 12 horses.

And since the total number of animals is 30, there must be 30 - 12 or 18 geese.

In conclusion, there are 12 horses and 18 geese.