Question 1131674
.
A mathematician turned farmer has cows and chickens. 
He tells you that among the cows and chickens, there are 148 legs and 60 heads. How many cows and chickens {{{highlight(are)}}} {{{highlight(there)}}} on his farm?
~~~~~~~~~~~~~~~~~~~


You can solve this problem in three different ways. 

If you are familiar with systems of linear equations,  you can reduce the problem to the system of two linear equations in two 

unknowns and solve it. &nbsp;This way is implemented in the &nbsp;<U>Solution 1</U>&nbsp; below.

You can also reduce the problem to one equation with one unknown and solve it. &nbsp;This is done  

in the &nbsp;<U>Solution 2</U>&nbsp; below.

Alternatively, &nbsp;you can solve the problem simply applying logical reasoning and not using equations at all. &nbsp;This is done 
in the &nbsp;<U>Solution 3</U>&nbsp; below.


<U>Solution 1</U>


Let &nbsp;<B>x</B>&nbsp; be the number of cows and &nbsp;<B>y</B>&nbsp; be the number of chicken at the farm.
If you count the heads you get the equation
x + y = 60.
If you count the legs you get the equation 
4x + 2y = 148.


So you have the system of two equations with two unknowns
{{{system (x + y = 60,
4x + 2y = 148)
}}}


To solve this system of equations, multiply the first equation by &nbsp;2&nbsp; and subtract the obtained equation from the second one. 
You will get, step by step,
{{{system (2x + 2y = 120,
4x + 2y = 148)
}}}


4x - 2x = 148 - 120,
2x = 28,
x = 14.


So, &nbsp;there are &nbsp;14&nbsp; cows in the farm. 
Hence, &nbsp;the number of chicken is &nbsp;60 - 14 = 46.


Let us check the total number of legs. &nbsp;You have altogether
4*14 + 2*46 = 148 legs.   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;! Correct !


<U>Answer</U>. &nbsp;There are &nbsp;14&nbsp; cows and &nbsp;46&nbsp; chicken at the farm.



<U>Solution 2</U>


Let &nbsp;<B>x</B>&nbsp; be the number of cows at a farm.
Then the number of chicken is &nbsp;60 - x&nbsp; in accordance with the condition.
If you count the legs you get the equation 
4x + 2*(60-x) = 148.


To solve this equation open the parentheses and combine like terms, &nbsp;step by step:
4x + 2*60 - 2x = 148,
2x + 120 = 148,
2x = 148 - 120,
2x = 28,
x = 14.


So, &nbsp;there are &nbsp;14&nbsp; cows at the farm. 
Hence, &nbsp;the number of chicken is &nbsp;60 - 14 = 46.


You got the same answer as in the <U>Solution 1</U>.



<U>Solution 3</U>


Let us suppose for a moment that all the animals at the farm have two legs each.
Under this assumption, the total number of legs is &nbsp;60*2 = 120 legs.


This number is in &nbsp;28&nbsp; less than &nbsp;148 &nbsp;legs given by condition. 
Certainly, &nbsp;these &nbsp;24&nbsp; legs belong to cows in the number of &nbsp;2&nbsp; legs to each cow &nbsp;(in addition to that two legs we just counted under the assumption). 
This means that the number of cows is &nbsp;28/2 = 14.
Hence, &nbsp;the number of chicken is &nbsp;60 - 14 = 46.


You get the same answer as in the solutions &nbsp;<B>1</B>&nbsp; and &nbsp;<B>2</B>&nbsp; above.


<U>Answer</U>. &nbsp;There are &nbsp;14&nbsp; cows and &nbsp;46&nbsp; chicken at the farm.



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