# Farmer Brown had ducks and cows. One day he noticed that the animals had a total of 16 heads and...

## Question:

Farmer Brown had ducks and cows. One day he noticed that the animals had a total of 16 heads and 42 feet.

How many of the animals were cows?

## Proportions:

Professions using proportions include chef and bakers. They use proportions to translate the amount of the ingredients when a bigger or smaller batch of a recipe is needed. Architects make plans and blueprints that uses proportions to depict the building to scale.

Let {eq}d {/eq} stand for the number of ducks and {eq}c {/eq} stand for the number of cows. Farmer brown notices that he had 16 animals, the equation for which would be:

{eq}d + c = 16 {/eq}

Identities to find an identity for finding ducks or cows, respectively, would be:

{eq}d = 16 -c {/eq} or {eq}c = 16 -d {/eq}

For the second equation using the number of feet:

{eq}2d + 4c = 42 {/eq}

Substituting the identity for the number of ducks in the second equation:

{eq}\begin{align} 2d + 4c &= 42 \\ 2(16 - c) + 4c &= 42 \\ 32 - 2c + 4c &= 42 \\ - 2c + 4c &= 42 - 32 \\ 2c &= 10 \\ c &= \dfrac{10}2 \\ c &= 5 \end{align} {/eq}

Substitutithe value to determine the number of ducks:

{eq}d = 16 -c \\ d = 16 - 5 \\ d = 11 {/eq}

This means that there are {eq}\rm 5 \ chickens {/eq} and {eq}\rm 11 \ ducks {/eq}. 