Math CentralQuandaries & Queries


Question from Kim, a parent:

Old McDonald raises goats and chickens. The animals have a total of 100 heads adn 360 feet.  How many goats and how many chickens does Mr. McDonald have?

Kim, we have two responses for you.

Hi Kim.

Let G = the number of goats and C = the number of chickens.

Each goat has one head and each chicken has one head and the total number of heads is 100, so:

1G + 1C = 100.

Each goat has two feet and each chicken has four feet and the total number of feet is 360, so:

2G + 4C = 360.

Now you have two questions with two unknown variables. It would be easy to use the elimination method to solve it from here (although you can also solve using the substitution method):

Elimination method:

First, look at the factors in front of the G variables to see if one is a multiple of the other (if not, try the C variables). In our case, 2 is a multiple of 1 because 2 x 1 = 2.

So we multiply the equation (both sides) be the factor we need to make it the same. In our case, this is 2.

2 ( 1G + 1C ) = 2 ( 100)                 which simplifies to

2G + 2C = 200.

So we have transformed the first equation from 1G + 1C = 100 into 2G + 2C = 200.

Now we write each equation below the other, lining up the variables:

2G +2C = 200
2G +4C = 360

And we subtract the bottom equation from the top (actually, we can just subtract the corresponding elements):

2G +2C = 200
2G +4C = 360
0G -2C = -160

Thus, we now have an equation with no G in it, which we can solve for C, the number of chickens. Then use that value of C in either of the original equations to find the corresponding number of goats (G).

-2C = -160.

Hope this helps,
Stephen La Rocque.


Hi Kim,

My granddaughter decided to make her own farmyard toys.

She has 100 marshmallows which she is going to use for heads and 360 toothpicks she will use for legs. She can put 4 toothpicks into a marshmallow to make a goat or 2 toothpicks into a marshmallow to make a chicken. She likes chickens so she takes 200 toothpicks and the 100 marshmallows and makes them into 100 chickens.

This leaves her with 160 unused legs or 80 pairs of unused legs. These she uses to turn some of the
chickens into goats.

How many goats and how many chickens does she build?


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