Question

in the barn there are ducks and pigs. Altogether there are 12 heads and 32 legs. How many ducks are in the barn?A) 8 ducksB) 4 ducksC) 6 ducksD) 7 ducks

Answer 1

Let be "d" the number of ducks in the barn and "p" the number of pigs in the barn.

According to the information given in the exercise, there are 12 heads and 32 legs.

Then:

- Assuming that each duck has 2 legs and each pig has 4 legs, you can set up this first equation:

`2d+4p=32`

- Since there are 12 heads, then the total number of ducks and pigs is 12. Knowing this, you can set up the second equation:

`d+p=12`

Now you can set up this System of Equations:

`\begin{cases}2d+4p=32 \\ d+p=12\end{cases}`

You can apply the Substitution Method in order to find the value of "d":

1. Choose the second equation and solve for "p":

`p=12-d`

2. Substitute this equation into the first original equation:

`2d+4(12-d)=32`

3. Solve for "d":

`\begin{gathered} 2d+(4)(12)-(4)(d)=32 \\ 2d+48-4d=32 \\ -2d+48=32 \\ -2d=32-48 \\ \\ d=(-16)/(-2) \\ \\ d=8 \end{gathered}`

Therefore, the answer is: Option A.