Question

Old McDonald raises chicken and sheep in his farm. His livestock has total of $55$ heads and $142$ legs among them (not counting of farmer). How many chickens and how many sheep does he have?

Answer 1

Step 1: Form the equations.

Let the number of chicken and sheep be $x$ and $y$ respectively.

Since each animal has one head, then the total number of heads of livestock in the farm is $x+y$.

It is given that, the total number of heads is $55$ heads.

$\therefore$ $x+y=55\dots \left(1\right)$

We know, that each chicken has $2$ legs and there are $x$ chickens.

So, total number of legs of chicken is $2x$.

Similarly, each sheep has $4$ legs and there are $y$ sheep.

So, total number of legs of sheep is $4y$.

Total number of legs of livestock is $2x+4y$.

It is given that there are $142$ legs in farm.

$\therefore 2x+4y=142\dots \left(2\right)$

Step 2: Solve the equations for $y$.

Multiplying equation $\left(1\right)$ by $2$ and subtracting it from equation $\left(2\right)$.

$\begin{array}{rcl}\left(2x+4y\right)-2\left(x+y\right)& =& 142-2·55\\ & \Rightarrow & 2x+4y-2x-2y=142-110\\ & \Rightarrow & 2y=32\\ & \Rightarrow & y=\frac{32}{2}\left(\mathrm{Dividing}\mathrm{both}\mathrm{sides}\mathrm{by}2\right)\\ \therefore y& =& 16\end{array}$

Step 3: Solve the equations for $x$.

Putting $y=16$ in equation $\left(1\right)$.

$\begin{array}{rcl}x+16& =& 55\\ & \Rightarrow & x=55-16\\ \therefore x& =& 39\end{array}$

Hence, number of chicken and sheep are $39$ and $16$ respectively.